Approximate the integral below using a left hand sum with 4 rectangles. $$ \int_{0}^{4}\left(16-x^{2}\right) \mathrm{dx} \approx[?] $$

Question

UpStudy AI · Accepted Answer

We are asked to approximate

$$
\int_{0}^{4}\left(16-x^{2}\right) \, dx
$$

using a left-hand sum with 4 rectangles.

**Step 1.** Calculate the width $$\Delta x$$ of each rectangle. Since the interval is $$[0, 4]$$ and we use 4 rectangles, we have

$$
\Delta x = \frac{4-0}{4} = 1.
$$

**Step 2.** Identify the left endpoints for each rectangle. They are

$$
x_0 = 0, \quad x_1 = 1, \quad x_2 = 2, \quad x_3 = 3.
$$

**Step 3.** Evaluate the function $$f(x) = 16 - x^2$$ at each left endpoint:

$$
f(x_0) = 16 - 0^2 = 16,
$$
$$
f(x_1) = 16 - 1^2 = 15,
$$
$$
f(x_2) = 16 - 2^2 = 16 - 4 = 12,
$$
$$
f(x_3) = 16 - 3^2 = 16 - 9 = 7.
$$

**Step 4.** Multiply each function value by the width $$\Delta x$$ and sum them up:

$$
\text{Left-hand sum } L = \Delta x \left(f(x_0) + f(x_1) + f(x_2) + f(x_3)\right)
$$
$$
L = 1 \left(16 + 15 + 12 + 7\right).
$$

**Step 5.** Compute the total:

$$
16 + 15 + 12 + 7 = 50.
$$

Thus, the approximate integral is

$$
\int_{0}^{4}\left(16-x^{2}\right) \, dx \approx 50.
$$
The approximate value of the integral is 50.

Approximate the integral below using a left hand sum with 4 rectangles. \[ \int_{0}^{4}\left(16-x^{2}\right) \mathrm{dx} \approx[?] \]