Chapter 5

Find the average value of each function over the given interval. \(f(x)=x^{2}\) on [0,3]

1-8. For each demand function \(d(x)\) and demand level \(x\) find the consumers' surplus. $$ d(x)=4000-12 x, \quad x=100 $$

1-34. Find each indefinite integral. \(\int e^{3 x} d x\)

Find each indefinite integral. \(\int x^{4} d x\)

Find each indefinite integral. [Integration formulas (A), (B), and (C) are on the inside back cover, numbered \(5-7]\) $$ \int\left(x^{3}+1\right)^{4} 3 x^{2} d x \quad\left[\begin{array}{l} \text { Hint: Use } u=x^{3}+1 \\ \text { and formula } 5 \end{array}\right] $$

Find each indefinite integral. \(\int e^{4 x} d x\)

For each demand function \(d(x)\) and demand level \(x\) find the consumers' surplus. $$ d(x)=500-x, \quad x=400 $$

Find the average value of each function over the given interval. \(f(x)=x^{3}\) on [0,2]

Find each indefinite integral. \(\int x^{7} d x\)

Find each indefinite integral. \(\int e^{x / 4} d x\)

For each demand function \(d(x)\) and demand level \(x\) find the consumers' surplus. $$ d(x)=300-\frac{1}{2} x, \quad x=200 $$

Find the average value of each function over the given interval. \(f(x)=3 \sqrt{x}\) on [0,4]

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

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

For each demand function \(d(x)\) and demand level \(x\) find the consumers' surplus. $$ d(x)=200-\frac{1}{2} x, \quad x=300 $$

Find the average value of each function over the given interval. \(f(x)=\sqrt[3]{x}\) on [0,8]

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

Find each indefinite integral. \(\int e^{0.05 x} d x\)

For each demand function \(d(x)\) and demand level \(x\) find the consumers' surplus. $$ d(x)=350-0.09 x^{2}, \quad x=50 $$

Find the average value of each function over the given interval. \(f(x)=\frac{1}{x^{2}}\) on [1,5]

Find each indefinite integral. \(\int \sqrt{u} d u\)

Find each indefinite integral. \(\int e^{0.02 x} d x\)

For each demand function \(d(x)\) and demand level \(x\) find the consumers' surplus. $$ d(x)=840-0.06 x^{2}, \quad x=100 $$

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

Find each indefinite integral. [Integration formulas (A), (B), and (C) are on the inside back cover, numbered \(5-7]\) $$ \int e^{x^{4}} x^{3} d x \quad\left[\begin{array}{l} \text { Hint: Use } u=x^{4} \\ \text { and formula } 7 . \end{array}\right] $$

Find each indefinite integral. \(\int \sqrt[3]{u} d u\)

Find each indefinite integral. \(\int e^{-2 y} d y\)

For each demand function \(d(x)\) and demand level \(x\) find the consumers' surplus. $$ d(x)=200 e^{-0.01 x}, x=100 $$

Find the average value of each function over the given interval. \(f(x)=2 x+1\) on [0,4]

Find each indefinite integral. \(\int \frac{d w}{w^{4}}\)

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

For each demand function \(d(x)\) and demand level \(x\) find the consumers' surplus. $$ d(x)=400 e^{-0.02 x}, x=75 $$

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

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

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

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

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

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 pages \(330-331,\) 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. $$ \begin{aligned} &f(x)=\sqrt{x} \text { from } a=1 \text { to } b=4\\\ &\text { For part (i), use } 6 \text { rectangles. } \end{aligned} $$

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

9-12. For each supply function \(s(x)\) and demand level \(x\), find the producers' surplus. $$ s(x)=0.02 x, \quad x=100 $$

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

Find each indefinite integral. \(\int e^{-0.4 x} d x\)

Find the average value of each function over the given interval. \(f(x)=9-x^{2}\) 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 pages \(330-331,\) 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. $$ \begin{aligned} &f(x)=e^{x} \text { from } a=-1 \text { to } b=1\\\ &\text { For part (i), use } 8 \text { rectangles. } \end{aligned} $$

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

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

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

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

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 pages \(330-331,\) 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. $$ \begin{aligned} &f(x)=\frac{1}{x} \text { from } a=1 \text { to } b=2\\\ &\text { For part (i), use } 10 \text { rectangles. } \end{aligned} $$

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