Chapter 11

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. $$ f(x)=3 x^{2}+1 \quad[-1,3] $$

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. $$ \int_{-4}^{0}\left[(x-6)-\left(x^{2}+5 x-6\right)\right] d x $$

Use the Exponential Rule to find the indefinite integral. $$ \int 3(x-4) e^{x^{2}-8 x} d x $$

Find the indefinite integral and check the result by differentiation. $$ \int\left(x^{2}-1\right)^{3}(2 x) d x $$

Find the indefinite integral and check your result by differentiation. $$ \int-4 d x $$

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. $$ f(x)=2 x-x^{3} $$

Sketch the region whose area is represented by the definite integral. Then use a geometric formula to evaluate the integral. $$ \int_{-3}^{3} \sqrt{9-x^{2}} d x $$

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

Find the indefinite integral and check the result by differentiation. $$ \int \sqrt{4 x^{2}-5}(8 x) d x $$

Find the indefinite integral and check your result by differentiation. $$ \int 5 t^{2} d t $$

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. $$ f(x)=x^{2}-x^{3} $$ $$ [0,1] $$

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. $$ \int_{-2}^{3}\left[(y+6)-y^{2}\right] d y $$

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

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

Find the indefinite integral and check the result by differentiation. $$ \int \sqrt[3]{1-2 x^{2}}(-4 x) d x $$

Find the indefinite integral and check your result by differentiation. $$ \int 3 t^{4} d t $$

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. $$ f(x)=x^{2}-x^{3} \quad[-1,0] $$

Determine which value best approximates the area of the region bounded by the graphs of \(f\) and \(g\). (Make your selection on the basis of a sketch of the region and not by performing any calculations.) \(f(x)=x+1, \quad g(x)=(x-1)^{2}\) (a) \(-2\) (b) 2 (c) 10 (d) 4 (e) \(\underline{8}\)

Use the values \(\int_{0}^{5} f(x) d x=6\) and \(\int_{0}^{5} g(x) d x=2\) to evaluate the definite integral. (a) \(\int_{0}^{5}[f(x)+g(x)] d x\) (b) \(\int_{0}^{5}[f(x)-g(x)] d x\) (c) \(\int_{0}^{5}-4 f(x) d x\) (d) \(\int_{0}^{5}[f(x)-3 g(x)] d x\)

Use the Log Rule to find the indefinite integral. $$ \int \frac{1}{x+1} d x $$

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

Find the indefinite integral and check your result by differentiation. $$ \int 5 x^{-3} d x $$

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. $$ f(x)=x(1-x)^{2} $$ $$ [0,1] $$

Determine which value best approximates the area of the region bounded by the graphs of \(f\) and \(g\). (Make your selection on the basis of a sketch of the region and not by performing any calculations.) \(f(x)=2-\frac{1}{2} x, \quad g(x)=2-\sqrt{x}\) (a) 1 (b) 6 (c) \(-3\) (d) 3 (e) 4

Use the values \(\int_{0}^{5} f(x) d x=6\) and \(\int_{0}^{5} g(x) d x=2\) to evaluate the definite integral. (a) \(\int_{0}^{5} 2 g(x) d x\) (b) \(\int_{5}^{0} f(x) d x\) (c) \(\int_{5}^{5} f(x) d x\) (d) \(\int_{0}^{5}[f(x)-f(x)] d x\)

Use the Log Rule to find the indefinite integral. $$ \int \frac{1}{x-5} d x $$

Find the indefinite integral and check the result by differentiation. $$ \int(x-3)^{5 / 2} d x $$

Find the indefinite integral and check your result by differentiation. $$ \int 4 y^{-3} d y $$

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. $$ f(x)=x^{2}(3-x) \quad[0,3] $$

Sketch the region bounded by the graphs of the functions and find the area of the region. $$ y=\frac{1}{x^{2}}, y=0, x=1, x=5 $$

Use the Log Rule to find the indefinite integral. $$ \int \frac{1}{3-2 x} d x $$

Find the indefinite integral and check the result by differentiation. $$ \int 2 x\left(x^{2}-1\right)^{7} d x $$

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

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. $$ f(x)=x^{2}+4 x \quad[0,4] $$

Sketch the region bounded by the graphs of the functions and find the area of the region. $$ y=x^{3}-2 x+1, y=-2 x, x=1 $$

Use the Log Rule to find the indefinite integral. $$ \int \frac{1}{6 x-5} d x $$

Find the indefinite integral and check the result by differentiation. $$ \int x\left(1-2 x^{2}\right)^{3} d $$

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

Sketch the region bounded by the graphs of the functions and find the area of the region. $$ f(x)=\sqrt[3]{x}, g(x)=x $$

Use the Log Rule to find the indefinite integral. $$ \int \frac{2}{3 x+5} d x $$

Find the indefinite integral and check the result by differentiation. $$ \int \frac{x^{2}}{\left(1+x^{3}\right)^{2}} d x $$

Find the indefinite integral and check your result by differentiation. $$ \int e d t $$

Use the Log Rule to find the indefinite integral. $$ \int \frac{5}{2 x-1} d x $$

Find the indefinite integral and check the result by differentiation. $$ \int \frac{x^{2}}{\left(x^{3}-1\right)^{2}} d x $$

Sketch the region bounded by the graphs of the functions and find the area of the region. $$ \begin{aligned} &y=x^{2}-4 x+3, y=3+4 x-x^{2} \\ &y=4-x^{2} \cdot y=x^{2} \end{aligned} $$

Use the Log Rule to find the indefinite integral. $$ \int \frac{x}{x^{2}+1} d x $$

Find the indefinite integral and check the result by differentiation. $$ \int \frac{x+1}{\left(x^{2}+2 x-3\right)^{2}} d x $$

Find the indefinite integral and check your result by differentiation. $$ \int y^{3 / 2} d y $$

Use the Log Rule to find the indefinite integral. $$ \int \frac{x^{2}}{3-x^{3}} d x $$

Find the indefinite integral and check the result by differentiation. $$ \int \frac{6 x}{\left(1+x^{2}\right)^{3}} d x $$