91Ó°ÊÓ

Find the average value of each function over the given interval. $$ f(x)=4 x-1 \text { on }[0,10] $$

Find each indefinite integral. \(\int e^{-3 y} d y\)

Find each indefinite integral. $$ \int \frac{d w}{w^{2}} $$

For each function: i. Approximate the area under the curve from \(a\) to \(b\) by calculating a Riemann sum with the given number of rectangles. Use the method described in Example 1 on page 351 , rounding to three decimal places. ii. Find the exact area under the curve from \(a\) to \(b\) by evaluating an appropriate definite integral using the Fundamental Theorem. \(f(x)=\sqrt{x}\) from \(a=1\) to \(b=4\) For part (i), use 6 rectangles.

Find each indefinite integral. \(\int e^{-0.5 x} d x\)

Find the average value of each function over the given interval. $$ f(x)=36-x^{2} \text { on }[-2,2] $$

Find each indefinite integral. $$ \int \frac{d z}{\sqrt{z}} $$

9-12. Show that each integral cannot be found by our substitution formulas. $$ \int \sqrt{x^{3}+1} x d x $$

For each function: i. Approximate the area under the curve from \(a\) to \(b\) by calculating a Riemann sum with the given number of rectangles. Use the method described in Example 1 on page 351 , rounding to three decimal places. ii. Find the exact area under the curve from \(a\) to \(b\) by evaluating an appropriate definite integral using the Fundamental Theorem. \(f(x)=e^{x}\) from \(a=-1\) to \(b=1\). For part (i), use 8 rectangles.

Find each indefinite integral. $$ \int \frac{d z}{\sqrt[3]{2}} $$