Chapter 11

Use the Midpoint Rule with \(n=4\) to approximate the area of the region. Compare your result with the exact area obtained with a definite integral. $$ f(x)=-2 x+3, \quad[0,1] $$

Find the area of the region. $$ \begin{aligned} &f(x)=x^{2}-6 x \\ &g(x)=0 \end{aligned} $$

Use a graphing utility to graph the integrand. Use the graph to determine whether the definite integral is positive, negative, or zero. $$ \begin{aligned} &\int_{0}^{3} \frac{5 x}{x^{2}+1} d x\\\ &\text { Exercises } 3-12, \text { sk } \end{aligned} $$

Use the Exponential Rule to find the indefinite integral. $$ \int 2 e^{2 x} d x $$

Identify \(u\) and \(d u / d x\) for the integral \(\int u^{n}(d u / d x) d x\). $$ \int\left(5 x^{2}+1\right)^{2}(10 x) d x $$

Verify the statement by showing that the derivative of the right side is equal to the integrand of the left side. $$ \int\left(-\frac{9}{x^{4}}\right) d x=\frac{3}{x^{3}}+C $$

Use the Midpoint Rule with \(n=4\) to approximate the area of the region. Compare your result with the exact area obtained with a definite integral. $$ f(x)=\frac{1}{x}, \quad[1,5] $$

Use a graphing utility to graph the integrand. Use the graph to determine whether the definite integral is positive, negative, or zero. $$ \int_{-2}^{2} x \sqrt{x^{2}+1} d x $$

Use the Exponential Rule to find the indefinite integral. $$ \int-3 e^{-3 x} d x $$

Identify \(u\) and \(d u / d x\) for the integral \(\int u^{n}(d u / d x) d x\). $$ \int\left(3-4 x^{2}\right)^{3}(-8 x) d x $$

Verify the statement by showing that the derivative of the right side is equal to the integrand of the left side. $$ \int \frac{4}{\sqrt{x}} d x=8 \sqrt{x}+C $$

Use the Midpoint Rule with \(n=4\) to approximate the area of the region. Compare your result with the exact area obtained with a definite integral. $$ f(x)=\sqrt{x}, \quad[0,1] $$

Sketch the region whose area is represented by the definite integral. Then use a geometric formula to evaluate the integral. $$ \int_{0} 3 d x $$

Use the Exponential Rule to find the indefinite integral. $$ \int e^{4 x} d x $$

Identify \(u\) and \(d u / d x\) for the integral \(\int u^{n}(d u / d x) d x\). $$ \int \sqrt{1-x^{2}}(-2 x) d x $$

Verify the statement by showing that the derivative of the right side is equal to the integrand of the left side. $$ \int\left(4 x^{3}-\frac{1}{x^{2}}\right) d x=x^{4}+\frac{1}{x}+C $$

Use the Midpoint Rule with \(n=4\) to approximate the area of the region. Compare your result with the exact area obtained with a definite integral. $$ f(x)=1-x^{2}, \quad[-1,1] $$

Sketch the region whose area is represented by the definite integral. Then use a geometric formula to evaluate the integral. $$ \int_{0}^{3} 4 d x $$

Use the Exponential Rule to find the indefinite integral. $$ \int e^{-0.25 x} d x $$

Identify \(u\) and \(d u / d x\) for the integral \(\int u^{n}(d u / d x) d x\). $$ \int 3 x^{2} \sqrt{x^{3}+1} d x $$

Verify the statement by showing that the derivative of the right side is equal to the integrand of the left side. $$ \int\left(1-\frac{1}{\sqrt[3]{x^{2}}}\right) d x=x-3 \sqrt[3]{x}+C $$

Use the Midpoint Rule with \(n=4\) to approximate the area of the region bounded by the graph of \(f\) and the \(x\) -axis over the interval. Compare your result with the exact area. Sketch the region. $$ f(x)=4-x^{2} $$ $$ [0,2] $$

Find the area of the region. $$ \begin{aligned} &f(x)=3\left(x^{3}-x\right) \\ &g(x)=0 \end{aligned} $$

Sketch the region whose area is represented by the definite integral. Then use a geometric formula to evaluate the integral. $$ \int_{0}^{4} x d x $$

Use the Exponential Rule to find the indefinite integral. $$ \int 9 x e^{-x^{2}} d x $$

Identify \(u\) and \(d u / d x\) for the integral \(\int u^{n}(d u / d x) d x\). $$ \int\left(4+\frac{1}{x^{2}}\right)^{5}\left(\frac{-2}{x^{3}}\right) d x $$

Verify the statement by showing that the derivative of the right side is equal to the integrand of the left side. $$ \int 2 \sqrt{x}(x-3) d x=\frac{4 x^{3 / 2}(x-5)}{5}+C $$

Use the Midpoint Rule with \(n=4\) to approximate the area of the region bounded by the graph of \(f\) and the \(x\) -axis over the interval. Compare your result with the exact area. Sketch the region. $$ f(x)=4 x^{2} $$ $$ [0,2] $$

Sketch the region whose area is represented by the definite integral. Then use a geometric formula to evaluate the integral. $$ \int_{0}^{4} \frac{x}{2} d x $$

Use the Exponential Rule to find the indefinite integral. $$ \int 3 x e^{0.5 x^{2}} d x $$

Identify \(u\) and \(d u / d x\) for the integral \(\int u^{n}(d u / d x) d x\). $$ \int \frac{1}{(1+2 x)^{2}}(2) d x $$

Verify the statement by showing that the derivative of the right side is equal to the integrand of the left side. $$ \int 4 \sqrt{x}\left(x^{2}-2\right) d x=\frac{8 x^{3 / 2}\left(3 x^{2}-14\right)}{21}+C $$

Use the Midpoint Rule with \(n=4\) to approximate the area of the region bounded by the graph of \(f\) and the \(x\) -axis over the interval. Compare your result with the exact area. Sketch the region. $$ f(x)=x^{2}+3 \quad[-1,1] $$

The integrand of the definite integral is a difference of two functions. Sketch the graph of each function and shade the region whose area is represented by the integral. $$ \int_{0}^{4}\left[(x+1)-\frac{1}{2} x\right] d x $$

Sketch the region whose area is represented by the definite integral. Then use a geometric formula to evaluate the integral. $$ \int_{0}^{5}(x+1) d x $$

Use the Exponential Rule to find the indefinite integral. $$ \int 5 x^{2} e^{x^{3}} d x $$

Identify \(u\) and \(d u / d x\) for the integral \(\int u^{n}(d u / d x) d x\). $$ \int(1+\sqrt{x})^{3}\left(\frac{1}{2 \sqrt{x}}\right) d x $$

Verify the statement by showing that the derivative of the right side is equal to the integrand of the left side. $$ \int(x-2)(x+2) d x=\frac{1}{3} x^{3}-4 x+C $$

Use the Midpoint Rule with \(n=4\) to approximate the area of the region bounded by the graph of \(f\) and the \(x\) -axis over the interval. Compare your result with the exact area. Sketch the region. $$ f(x)=4-x^{2} \quad[-2,2] $$

The integrand of the definite integral is a difference of two functions. Sketch the graph of each function and shade the region whose area is represented by the integral. $$ \int_{-1}^{1}\left[\left(1-x^{2}\right)-\left(x^{2}-1\right)\right] d x $$

Sketch the region whose area is represented by the definite integral. Then use a geometric formula to evaluate the integral. $$ \int_{0}^{3}(2 x+1) d x $$

Use the Exponential Rule to find the indefinite integral. $$ \int(2 x+1) e^{x^{2}+x} d x $$

Identify \(u\) and \(d u / d x\) for the integral \(\int u^{n}(d u / d x) d x\). $$ \int(4-\sqrt{x})^{2}\left(\frac{-1}{2 \sqrt{x}}\right) d x $$

Verify the statement by showing that the derivative of the right side is equal to the integrand of the left side. $$ \int \frac{x^{2}-1}{x^{3 / 2}} d x=\frac{2\left(x^{2}+3\right)}{3 \sqrt{x}}+C $$

Use the Midpoint Rule with \(n=4\) to approximate the area of the region bounded by the graph of \(f\) and the \(x\) -axis over the interval. Compare your result with the exact area. Sketch the region. $$ f(x)=2 x^{2} $$

The integrand of the definite integral is a difference of two functions. Sketch the graph of each function and shade the region whose area is represented by the integral. $$ \int_{-2}^{2}\left[2 x^{2}-\left(x^{4}-2 x^{2}\right)\right] d x $$

Sketch the region whose area is represented by the definite integral. Then use a geometric formula to evaluate the integral. $$ \int_{-2}^{3}|x-1| d x $$

Use the Exponential Rule to find the indefinite integral. $$ \int\left(x^{2}+2 x\right) e^{x^{3}+3 x^{2}-1} d x $$

Find the indefinite integral and check the result by differentiation. $$ \int(1+2 x)^{4}(2) d x $$

Find the indefinite integral and check your result by differentiation. $$ \int 6 d x $$