During the time interval \( 0 \leq t \leq 6 \), the rate at which the amount of water in a tank is changing, in liters per minute, can be modeled by the function \( E(t)=49 \sin \left(0.11 t^{2}\right)+17 \), where \( t \) is measured in minutes. What is the average rate of change of the function \( E(t) \) from time \( t=0 \) to time \( t=6 \) ? You may use a calculator and round to the nearest thousandth. Indicate units of measure.

Use one or more definite integrals to find the area between the \( x \)-axis and the given curve on the given interval. Be sure to find if the curve crosses the \( x \)-axis in the given interval. \( f(x)=3-3 x^{2} \) on \( [0,2] \) Area:

Use one or more definite integrals to find the area between the \( x \)-axis and the given curve on the given interval. Be sure to find if the curve crosses the \( x \)-axis in the given interval. \( f(x)=4 e^{x}-4 \) on \( [-2,2] \) Area:

Evaluate the expression \( \int_{3}^{5} x^{2} \sin \sqrt{x} \ln x d x+\int_{5}^{3} x^{2} \sin \sqrt{x} \ln x d x \) Answer: \( \square \)

\( \int _ { 0 } ^ { 1 } \int _ { 0 } ^ { \sqrt { 1 - x ^ { 2 } } } \sqrt { 1 - x ^ { 2 } - y ^ { 2 } } d y d x \)

\( \iint_{R} d A \), si \( R \) es la región, dada por: \[ R=\left\{(x, y) \in \mathbb{R}^{2}, 0 \leq x \leq \frac{1}{2}, y+x \leq 1, y \geq 0 .\right\} \]