Approximate the integral below using
a left hand sum with 6 rectangles.
Round your final answer to the
nearest thousandth.

Ask by Reid Beck. in the United States

Mar 22,2025

Question

Approximate the integral below using
a left hand sum with 6 rectangles.
Round your final answer to the
nearest thousandth.

Ask by Reid Beck. in the United States

Mar 22,2025

UpStudy AI · Accepted Answer

Calculate or simplify the expression $$ \int_{1}^{4}(\frac{3}{2}\sqrt{x}-\frac{2}{x})dx $$.
Evaluate the integral by following steps:
- step0: Evaluate using formulas and rules:
  $$\int_{1}^{4} \left(\frac{3}{2}\sqrt{x}-\frac{2}{x}ight) dx$$
- step1: Simplify:
  $$\int_{1}^{4} \left(\frac{3}{2}x^{\frac{1}{2}}-\frac{2}{x}ight) dx$$
- step2: Evaluate the integral:
  $$\int \left(\frac{3}{2}x^{\frac{1}{2}}-\frac{2}{x}ight) dx$$
- step3: Use properties of integrals:
  $$\int \frac{3}{2}x^{\frac{1}{2}} dx+\int -\frac{2}{x} dx$$
- step4: Evaluate the integral:
  $$x^{\frac{3}{2}}+\int -\frac{2}{x} dx$$
- step5: Evaluate the integral:
  $$x^{\frac{3}{2}}-2\ln{\left(\left|xight|ight)}$$
- step6: Return the limits:
  $$\left(x^{\frac{3}{2}}-2\ln{\left(\left|xight|ight)}ight)\bigg |_{1}^{4}$$
- step7: Calculate the value:
  $$7-2	imes 2\ln{\left(2ight)}$$
- step8: Simplify:
  $$7-4\ln{\left(2ight)}$$
To approximate the integral using a left hand sum with 6 rectangles, we need to divide the interval $$[1, 4]$$ into 6 equal subintervals. The width of each subinterval is $$\frac{4-1}{6} = \frac{3}{6} = \frac{1}{2}$$.

The left hand sum is calculated by multiplying the function value at the left endpoint of each subinterval by the width of the subinterval and summing these values.

Let's calculate the left hand sum for the given integral.
Calculate the value by following steps:
- step0: Calculate:
  $$\left(\frac{3}{2}\sqrt{1}-\frac{2}{1}ight)	imes \frac{1}{2}+\left(\frac{3}{2}\sqrt{1.5}-\frac{2}{1.5}ight)	imes \frac{1}{2}+\left(\frac{3}{2}\sqrt{2}-\frac{2}{2}ight)	imes \frac{1}{2}+\left(\frac{3}{2}\sqrt{2.5}-\frac{2}{2.5}ight)	imes \frac{1}{2}+\left(\frac{3}{2}\sqrt{3}-\frac{2}{3}ight)	imes \frac{1}{2}+\left(\frac{3}{2}\sqrt{3.5}-\frac{2}{3.5}ight)	imes \frac{1}{2}$$
- step1: Divide the terms:
  $$\left(\frac{3}{2}\sqrt{1}-\frac{2}{1}ight)	imes \frac{1}{2}+\left(\frac{3}{2}\sqrt{1.5}-\frac{2}{1.5}ight)	imes \frac{1}{2}+\left(\frac{3}{2}\sqrt{2}-1ight)	imes \frac{1}{2}+\left(\frac{3}{2}\sqrt{2.5}-\frac{2}{2.5}ight)	imes \frac{1}{2}+\left(\frac{3}{2}\sqrt{3}-\frac{2}{3}ight)	imes \frac{1}{2}+\left(\frac{3}{2}\sqrt{3.5}-\frac{2}{3.5}ight)	imes \frac{1}{2}$$
- step2: Simplify the root:
  $$\left(\frac{3}{2}	imes 1-\frac{2}{1}ight)	imes \frac{1}{2}+\left(\frac{3}{2}\sqrt{1.5}-\frac{2}{1.5}ight)	imes \frac{1}{2}+\left(\frac{3}{2}\sqrt{2}-1ight)	imes \frac{1}{2}+\left(\frac{3}{2}\sqrt{2.5}-\frac{2}{2.5}ight)	imes \frac{1}{2}+\left(\frac{3}{2}\sqrt{3}-\frac{2}{3}ight)	imes \frac{1}{2}+\left(\frac{3}{2}\sqrt{3.5}-\frac{2}{3.5}ight)	imes \frac{1}{2}$$
- step3: Divide the terms:
  $$\left(\frac{3}{2}	imes 1-2ight)	imes \frac{1}{2}+\left(\frac{3}{2}\sqrt{1.5}-\frac{2}{1.5}ight)	imes \frac{1}{2}+\left(\frac{3}{2}\sqrt{2}-1ight)	imes \frac{1}{2}+\left(\frac{3}{2}\sqrt{2.5}-\frac{2}{2.5}ight)	imes \frac{1}{2}+\left(\frac{3}{2}\sqrt{3}-\frac{2}{3}ight)	imes \frac{1}{2}+\left(\frac{3}{2}\sqrt{3.5}-\frac{2}{3.5}ight)	imes \frac{1}{2}$$
- step4: Multiply:
  $$\left(\frac{3}{2}-2ight)	imes \frac{1}{2}+\left(\frac{3}{2}\sqrt{1.5}-\frac{2}{1.5}ight)	imes \frac{1}{2}+\left(\frac{3}{2}\sqrt{2}-1ight)	imes \frac{1}{2}+\left(\frac{3}{2}\sqrt{2.5}-\frac{2}{2.5}ight)	imes \frac{1}{2}+\left(\frac{3}{2}\sqrt{3}-\frac{2}{3}ight)	imes \frac{1}{2}+\left(\frac{3}{2}\sqrt{3.5}-\frac{2}{3.5}ight)	imes \frac{1}{2}$$
- step5: Subtract the numbers:
  $$\left(-\frac{1}{2}ight)	imes \frac{1}{2}+\left(\frac{3}{2}\sqrt{1.5}-\frac{2}{1.5}ight)	imes \frac{1}{2}+\left(\frac{3}{2}\sqrt{2}-1ight)	imes \frac{1}{2}+\left(\frac{3}{2}\sqrt{2.5}-\frac{2}{2.5}ight)	imes \frac{1}{2}+\left(\frac{3}{2}\sqrt{3}-\frac{2}{3}ight)	imes \frac{1}{2}+\left(\frac{3}{2}\sqrt{3.5}-\frac{2}{3.5}ight)	imes \frac{1}{2}$$
- step6: Remove the parentheses:
  $$-\frac{1}{2}	imes \frac{1}{2}+\left(\frac{3}{2}\sqrt{1.5}-\frac{2}{1.5}ight)	imes \frac{1}{2}+\left(\frac{3}{2}\sqrt{2}-1ight)	imes \frac{1}{2}+\left(\frac{3}{2}\sqrt{2.5}-\frac{2}{2.5}ight)	imes \frac{1}{2}+\left(\frac{3}{2}\sqrt{3}-\frac{2}{3}ight)	imes \frac{1}{2}+\left(\frac{3}{2}\sqrt{3.5}-\frac{2}{3.5}ight)	imes \frac{1}{2}$$
- step7: Simplify the root:
  $$-\frac{1}{2}	imes \frac{1}{2}+\left(\frac{3}{2}	imes \frac{\sqrt{6}}{2}-\frac{2}{1.5}ight)	imes \frac{1}{2}+\left(\frac{3}{2}\sqrt{2}-1ight)	imes \frac{1}{2}+\left(\frac{3}{2}\sqrt{2.5}-\frac{2}{2.5}ight)	imes \frac{1}{2}+\left(\frac{3}{2}\sqrt{3}-\frac{2}{3}ight)	imes \frac{1}{2}+\left(\frac{3}{2}\sqrt{3.5}-\frac{2}{3.5}ight)	imes \frac{1}{2}$$
- step8: Divide the numbers:
  $$-\frac{1}{2}	imes \frac{1}{2}+\left(\frac{3}{2}	imes \frac{\sqrt{6}}{2}-\frac{4}{3}ight)	imes \frac{1}{2}+\left(\frac{3}{2}\sqrt{2}-1ight)	imes \frac{1}{2}+\left(\frac{3}{2}\sqrt{2.5}-\frac{2}{2.5}ight)	imes \frac{1}{2}+\left(\frac{3}{2}\sqrt{3}-\frac{2}{3}ight)	imes \frac{1}{2}+\left(\frac{3}{2}\sqrt{3.5}-\frac{2}{3.5}ight)	imes \frac{1}{2}$$
- step9: Multiply the numbers:
  $$-\frac{1}{2}	imes \frac{1}{2}+\left(\frac{3\sqrt{6}}{4}-\frac{4}{3}ight)	imes \frac{1}{2}+\left(\frac{3}{2}\sqrt{2}-1ight)	imes \frac{1}{2}+\left(\frac{3}{2}\sqrt{2.5}-\frac{2}{2.5}ight)	imes \frac{1}{2}+\left(\frac{3}{2}\sqrt{3}-\frac{2}{3}ight)	imes \frac{1}{2}+\left(\frac{3}{2}\sqrt{3.5}-\frac{2}{3.5}ight)	imes \frac{1}{2}$$
- step10: Subtract the numbers:
  $$-\frac{1}{2}	imes \frac{1}{2}+\frac{9\sqrt{6}-16}{12}	imes \frac{1}{2}+\left(\frac{3}{2}\sqrt{2}-1ight)	imes \frac{1}{2}+\left(\frac{3}{2}\sqrt{2.5}-\frac{2}{2.5}ight)	imes \frac{1}{2}+\left(\frac{3}{2}\sqrt{3}-\frac{2}{3}ight)	imes \frac{1}{2}+\left(\frac{3}{2}\sqrt{3.5}-\frac{2}{3.5}ight)	imes \frac{1}{2}$$
- step11: Multiply:
  $$-\frac{1}{2}	imes \frac{1}{2}+\frac{9\sqrt{6}-16}{12}	imes \frac{1}{2}+\left(\frac{3\sqrt{2}}{2}-1ight)	imes \frac{1}{2}+\left(\frac{3}{2}\sqrt{2.5}-\frac{2}{2.5}ight)	imes \frac{1}{2}+\left(\frac{3}{2}\sqrt{3}-\frac{2}{3}ight)	imes \frac{1}{2}+\left(\frac{3}{2}\sqrt{3.5}-\frac{2}{3.5}ight)	imes \frac{1}{2}$$
- step12: Subtract the numbers:
  $$-\frac{1}{2}	imes \frac{1}{2}+\frac{9\sqrt{6}-16}{12}	imes \frac{1}{2}+\frac{3\sqrt{2}-2}{2}	imes \frac{1}{2}+\left(\frac{3}{2}\sqrt{2.5}-\frac{2}{2.5}ight)	imes \frac{1}{2}+\left(\frac{3}{2}\sqrt{3}-\frac{2}{3}ight)	imes \frac{1}{2}+\left(\frac{3}{2}\sqrt{3.5}-\frac{2}{3.5}ight)	imes \frac{1}{2}$$
- step13: Simplify the root:
  $$-\frac{1}{2}	imes \frac{1}{2}+\frac{9\sqrt{6}-16}{12}	imes \frac{1}{2}+\frac{3\sqrt{2}-2}{2}	imes \frac{1}{2}+\left(\frac{3}{2}	imes \frac{\sqrt{10}}{2}-\frac{2}{2.5}ight)	imes \frac{1}{2}+\left(\frac{3}{2}\sqrt{3}-\frac{2}{3}ight)	imes \frac{1}{2}+\left(\frac{3}{2}\sqrt{3.5}-\frac{2}{3.5}ight)	imes \frac{1}{2}$$
- step14: Divide the numbers:
  $$-\frac{1}{2}	imes \frac{1}{2}+\frac{9\sqrt{6}-16}{12}	imes \frac{1}{2}+\frac{3\sqrt{2}-2}{2}	imes \frac{1}{2}+\left(\frac{3}{2}	imes \frac{\sqrt{10}}{2}-\frac{4}{5}ight)	imes \frac{1}{2}+\left(\frac{3}{2}\sqrt{3}-\frac{2}{3}ight)	imes \frac{1}{2}+\left(\frac{3}{2}\sqrt{3.5}-\frac{2}{3.5}ight)	imes \frac{1}{2}$$
- step15: Multiply the numbers:
  $$-\frac{1}{2}	imes \frac{1}{2}+\frac{9\sqrt{6}-16}{12}	imes \frac{1}{2}+\frac{3\sqrt{2}-2}{2}	imes \frac{1}{2}+\left(\frac{3\sqrt{10}}{4}-\frac{4}{5}ight)	imes \frac{1}{2}+\left(\frac{3}{2}\sqrt{3}-\frac{2}{3}ight)	imes \frac{1}{2}+\left(\frac{3}{2}\sqrt{3.5}-\frac{2}{3.5}ight)	imes \frac{1}{2}$$
- step16: Subtract the numbers:
  $$-\frac{1}{2}	imes \frac{1}{2}+\frac{9\sqrt{6}-16}{12}	imes \frac{1}{2}+\frac{3\sqrt{2}-2}{2}	imes \frac{1}{2}+\frac{15\sqrt{10}-16}{20}	imes \frac{1}{2}+\left(\frac{3}{2}\sqrt{3}-\frac{2}{3}ight)	imes \frac{1}{2}+\left(\frac{3}{2}\sqrt{3.5}-\frac{2}{3.5}ight)	imes \frac{1}{2}$$
- step17: Multiply:
  $$-\frac{1}{2}	imes \frac{1}{2}+\frac{9\sqrt{6}-16}{12}	imes \frac{1}{2}+\frac{3\sqrt{2}-2}{2}	imes \frac{1}{2}+\frac{15\sqrt{10}-16}{20}	imes \frac{1}{2}+\left(\frac{3\sqrt{3}}{2}-\frac{2}{3}ight)	imes \frac{1}{2}+\left(\frac{3}{2}\sqrt{3.5}-\frac{2}{3.5}ight)	imes \frac{1}{2}$$
- step18: Subtract the numbers:
  $$-\frac{1}{2}	imes \frac{1}{2}+\frac{9\sqrt{6}-16}{12}	imes \frac{1}{2}+\frac{3\sqrt{2}-2}{2}	imes \frac{1}{2}+\frac{15\sqrt{10}-16}{20}	imes \frac{1}{2}+\frac{9\sqrt{3}-4}{6}	imes \frac{1}{2}+\left(\frac{3}{2}\sqrt{3.5}-\frac{2}{3.5}ight)	imes \frac{1}{2}$$
- step19: Simplify the root:
  $$-\frac{1}{2}	imes \frac{1}{2}+\frac{9\sqrt{6}-16}{12}	imes \frac{1}{2}+\frac{3\sqrt{2}-2}{2}	imes \frac{1}{2}+\frac{15\sqrt{10}-16}{20}	imes \frac{1}{2}+\frac{9\sqrt{3}-4}{6}	imes \frac{1}{2}+\left(\frac{3}{2}	imes \frac{\sqrt{14}}{2}-\frac{2}{3.5}ight)	imes \frac{1}{2}$$
- step20: Divide the numbers:
  $$-\frac{1}{2}	imes \frac{1}{2}+\frac{9\sqrt{6}-16}{12}	imes \frac{1}{2}+\frac{3\sqrt{2}-2}{2}	imes \frac{1}{2}+\frac{15\sqrt{10}-16}{20}	imes \frac{1}{2}+\frac{9\sqrt{3}-4}{6}	imes \frac{1}{2}+\left(\frac{3}{2}	imes \frac{\sqrt{14}}{2}-\frac{4}{7}ight)	imes \frac{1}{2}$$
- step21: Multiply the numbers:
  $$-\frac{1}{2}	imes \frac{1}{2}+\frac{9\sqrt{6}-16}{12}	imes \frac{1}{2}+\frac{3\sqrt{2}-2}{2}	imes \frac{1}{2}+\frac{15\sqrt{10}-16}{20}	imes \frac{1}{2}+\frac{9\sqrt{3}-4}{6}	imes \frac{1}{2}+\left(\frac{3\sqrt{14}}{4}-\frac{4}{7}ight)	imes \frac{1}{2}$$
- step22: Subtract the numbers:
  $$-\frac{1}{2}	imes \frac{1}{2}+\frac{9\sqrt{6}-16}{12}	imes \frac{1}{2}+\frac{3\sqrt{2}-2}{2}	imes \frac{1}{2}+\frac{15\sqrt{10}-16}{20}	imes \frac{1}{2}+\frac{9\sqrt{3}-4}{6}	imes \frac{1}{2}+\frac{21\sqrt{14}-16}{28}	imes \frac{1}{2}$$
- step23: Multiply the numbers:
  $$-\frac{1}{4}+\frac{9\sqrt{6}-16}{12}	imes \frac{1}{2}+\frac{3\sqrt{2}-2}{2}	imes \frac{1}{2}+\frac{15\sqrt{10}-16}{20}	imes \frac{1}{2}+\frac{9\sqrt{3}-4}{6}	imes \frac{1}{2}+\frac{21\sqrt{14}-16}{28}	imes \frac{1}{2}$$
- step24: Multiply the numbers:
  $$-\frac{1}{4}+\frac{9\sqrt{6}-16}{24}+\frac{3\sqrt{2}-2}{2}	imes \frac{1}{2}+\frac{15\sqrt{10}-16}{20}	imes \frac{1}{2}+\frac{9\sqrt{3}-4}{6}	imes \frac{1}{2}+\frac{21\sqrt{14}-16}{28}	imes \frac{1}{2}$$
- step25: Multiply the numbers:
  $$-\frac{1}{4}+\frac{9\sqrt{6}-16}{24}+\frac{3\sqrt{2}-2}{4}+\frac{15\sqrt{10}-16}{20}	imes \frac{1}{2}+\frac{9\sqrt{3}-4}{6}	imes \frac{1}{2}+\frac{21\sqrt{14}-16}{28}	imes \frac{1}{2}$$
- step26: Multiply the numbers:
  $$-\frac{1}{4}+\frac{9\sqrt{6}-16}{24}+\frac{3\sqrt{2}-2}{4}+\frac{15\sqrt{10}-16}{40}+\frac{9\sqrt{3}-4}{6}	imes \frac{1}{2}+\frac{21\sqrt{14}-16}{28}	imes \frac{1}{2}$$
- step27: Multiply the numbers:
  $$-\frac{1}{4}+\frac{9\sqrt{6}-16}{24}+\frac{3\sqrt{2}-2}{4}+\frac{15\sqrt{10}-16}{40}+\frac{9\sqrt{3}-4}{12}+\frac{21\sqrt{14}-16}{28}	imes \frac{1}{2}$$
- step28: Multiply the numbers:
  $$-\frac{1}{4}+\frac{9\sqrt{6}-16}{24}+\frac{3\sqrt{2}-2}{4}+\frac{15\sqrt{10}-16}{40}+\frac{9\sqrt{3}-4}{12}+\frac{21\sqrt{14}-16}{56}$$
- step29: Reduce fractions to a common denominator:
  $$-\frac{6	imes 5	imes 7}{4	imes 6	imes 5	imes 7}+\frac{\left(9\sqrt{6}-16ight)	imes 5	imes 7}{24	imes 5	imes 7}+\frac{\left(3\sqrt{2}-2ight)	imes 6	imes 5	imes 7}{4	imes 6	imes 5	imes 7}+\frac{\left(15\sqrt{10}-16ight)	imes 3	imes 7}{40	imes 3	imes 7}+\frac{\left(9\sqrt{3}-4ight)	imes 10	imes 7}{12	imes 10	imes 7}+\frac{\left(21\sqrt{14}-16ight)	imes 15}{56	imes 15}$$
- step30: Multiply the terms:
  $$-\frac{6	imes 5	imes 7}{840}+\frac{\left(9\sqrt{6}-16ight)	imes 5	imes 7}{24	imes 5	imes 7}+\frac{\left(3\sqrt{2}-2ight)	imes 6	imes 5	imes 7}{4	imes 6	imes 5	imes 7}+\frac{\left(15\sqrt{10}-16ight)	imes 3	imes 7}{40	imes 3	imes 7}+\frac{\left(9\sqrt{3}-4ight)	imes 10	imes 7}{12	imes 10	imes 7}+\frac{\left(21\sqrt{14}-16ight)	imes 15}{56	imes 15}$$
- step31: Multiply the terms:
  $$-\frac{6	imes 5	imes 7}{840}+\frac{\left(9\sqrt{6}-16ight)	imes 5	imes 7}{840}+\frac{\left(3\sqrt{2}-2ight)	imes 6	imes 5	imes 7}{4	imes 6	imes 5	imes 7}+\frac{\left(15\sqrt{10}-16ight)	imes 3	imes 7}{40	imes 3	imes 7}+\frac{\left(9\sqrt{3}-4ight)	imes 10	imes 7}{12	imes 10	imes 7}+\frac{\left(21\sqrt{14}-16ight)	imes 15}{56	imes 15}$$
- step32: Multiply the terms:
  $$-\frac{6	imes 5	imes 7}{840}+\frac{\left(9\sqrt{6}-16ight)	imes 5	imes 7}{840}+\frac{\left(3\sqrt{2}-2ight)	imes 6	imes 5	imes 7}{840}+\frac{\left(15\sqrt{10}-16ight)	imes 3	imes 7}{40	imes 3	imes 7}+\frac{\left(9\sqrt{3}-4ight)	imes 10	imes 7}{12	imes 10	imes 7}+\frac{\left(21\sqrt{14}-16ight)	imes 15}{56	imes 15}$$
- step33: Multiply the terms:
  $$-\frac{6	imes 5	imes 7}{840}+\frac{\left(9\sqrt{6}-16ight)	imes 5	imes 7}{840}+\frac{\left(3\sqrt{2}-2ight)	imes 6	imes 5	imes 7}{840}+\frac{\left(15\sqrt{10}-16ight)	imes 3	imes 7}{840}+\frac{\left(9\sqrt{3}-4ight)	imes 10	imes 7}{12	imes 10	imes 7}+\frac{\left(21\sqrt{14}-16ight)	imes 15}{56	imes 15}$$
- step34: Multiply the terms:
  $$-\frac{6	imes 5	imes 7}{840}+\frac{\left(9\sqrt{6}-16ight)	imes 5	imes 7}{840}+\frac{\left(3\sqrt{2}-2ight)	imes 6	imes 5	imes 7}{840}+\frac{\left(15\sqrt{10}-16ight)	imes 3	imes 7}{840}+\frac{\left(9\sqrt{3}-4ight)	imes 10	imes 7}{840}+\frac{\left(21\sqrt{14}-16ight)	imes 15}{56	imes 15}$$
- step35: Multiply the numbers:
  $$-\frac{6	imes 5	imes 7}{840}+\frac{\left(9\sqrt{6}-16ight)	imes 5	imes 7}{840}+\frac{\left(3\sqrt{2}-2ight)	imes 6	imes 5	imes 7}{840}+\frac{\left(15\sqrt{10}-16ight)	imes 3	imes 7}{840}+\frac{\left(9\sqrt{3}-4ight)	imes 10	imes 7}{840}+\frac{\left(21\sqrt{14}-16ight)	imes 15}{840}$$
- step36: Transform the expression:
  $$\frac{-6	imes 5	imes 7+\left(9\sqrt{6}-16ight)	imes 5	imes 7+\left(3\sqrt{2}-2ight)	imes 6	imes 5	imes 7+\left(15\sqrt{10}-16ight)	imes 3	imes 7+\left(9\sqrt{3}-4ight)	imes 10	imes 7+\left(21\sqrt{14}-16ight)	imes 15}{840}$$
- step37: Multiply the terms:
  $$\frac{-210+\left(9\sqrt{6}-16ight)	imes 5	imes 7+\left(3\sqrt{2}-2ight)	imes 6	imes 5	imes 7+\left(15\sqrt{10}-16ight)	imes 3	imes 7+\left(9\sqrt{3}-4ight)	imes 10	imes 7+\left(21\sqrt{14}-16ight)	imes 15}{840}$$
- step38: Multiply the terms:
  $$\frac{-210+315\sqrt{6}-560+\left(3\sqrt{2}-2ight)	imes 6	imes 5	imes 7+\left(15\sqrt{10}-16ight)	imes 3	imes 7+\left(9\sqrt{3}-4ight)	imes 10	imes 7+\left(21\sqrt{14}-16ight)	imes 15}{840}$$
- step39: Multiply the terms:
  $$\frac{-210+315\sqrt{6}-560+630\sqrt{2}-420+\left(15\sqrt{10}-16ight)	imes 3	imes 7+\left(9\sqrt{3}-4ight)	imes 10	imes 7+\left(21\sqrt{14}-16ight)	imes 15}{840}$$
- step40: Multiply the terms:
  $$\frac{-210+315\sqrt{6}-560+630\sqrt{2}-420+315\sqrt{10}-336+\left(9\sqrt{3}-4ight)	imes 10	imes 7+\left(21\sqrt{14}-16ight)	imes 15}{840}$$
- step41: Multiply the terms:
  $$\frac{-210+315\sqrt{6}-560+630\sqrt{2}-420+315\sqrt{10}-336+630\sqrt{3}-280+\left(21\sqrt{14}-16ight)	imes 15}{840}$$
- step42: Multiply the terms:
  $$\frac{-210+315\sqrt{6}-560+630\sqrt{2}-420+315\sqrt{10}-336+630\sqrt{3}-280+315\sqrt{14}-240}{840}$$
- step43: Subtract the numbers:
  $$\frac{-2046+315\sqrt{6}+630\sqrt{2}+315\sqrt{10}+630\sqrt{3}+315\sqrt{14}}{840}$$
- step44: Rewrite the expression:
  $$\frac{3\left(-682+105\sqrt{6}+210\sqrt{2}+105\sqrt{10}+210\sqrt{3}+105\sqrt{14}ight)}{840}$$
- step45: Reduce the fraction:
  $$\frac{-682+105\sqrt{6}+210\sqrt{2}+105\sqrt{10}+210\sqrt{3}+105\sqrt{14}}{280}$$
The left hand sum approximation of the integral $$\int_{1}^{4}\left(\frac{3}{2} \sqrt{x}-\frac{2}{x}ight) dx$$ using 6 rectangles is approximately 3.431518.
The left-hand sum approximation of the integral is approximately 3.432.

Approximate the integral below using
a left hand sum with 6 rectangles.
Round your final answer to the
nearest thousandth.

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Approximate the integral below using a left hand sum with 6 rectangles. Round your final answer to the nearest thousandth.

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Approximate the integral below using
a left hand sum with 6 rectangles.
Round your final answer to the
nearest thousandth.