91Ó°ÊÓ

Show that each integral cannot be found by our substitution formulas. $$ \int \sqrt{x^{5}+9} x^{2} d x $$

Find the average value of each function over the given interval. $$ f(x)=9-x^{2} \text { on }[-3,3] $$

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.

For each supply function \(s(x)\) and demand level \(x\) find the producers' surplus. $$ s(x)=0.03 x^{2}, \quad x=200 $$

Find each indefinite integral. $$ \int 6 x^{5} d x $$

Show that each integral cannot be found by our substitution formulas. $$ \int e^{x^{4}} x^{5} d x $$

Find the average value of each function over the given interval. $$ f(z)=3 z^{2}-2 z \text { on }[-1,2] $$

Find each indefinite integral. \(\int 6 e^{2 x / 3} d x\)

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. \(f(x)=\frac{1}{\sqrt{x}}\) from \(a=1\) to \(b=4\). For part (i), use 6 rectangles.

Show that each integral cannot be found by our substitution formulas. $$ \int e^{x^{3}} x^{4} d x $$