Chapter 6

Find the volumes of the solids obtained by rotating the region bounded by the given curves about the \(x\) -axis. In each case, sketch the region and a typical disk element. \(y=\sqrt{2 x}, y=0, x=2\)

Approximate $$\int_{-1}^{2} e^{-x} d x$$ using three equal subintervals and midpoints.

Use Leibniz's rule to find \(\frac{d y}{d x}\). $$ y=\int_{2-x^{2}}^{x+x^{3}} \sin t d t $$

Find the volumes of the solids obtained by rotating the region bounded by the given curves about the \(x\) -axis. In each case, sketch the region and a typical disk element. \(y=\sqrt{\sin x}, 0 \leq x \leq \pi, y=0\)

Approximate $$\int_{0}^{3 \pi / 2} \sin x d x$$ using three equal subintervals and right endpoints.

Use Leibniz's rule to find \(\frac{d y}{d x}\). $$ y=\int_{1 \neq x^{2}}^{x^{3}-2 x} \cos t d t $$

Find the volumes of the solids obtained by rotating the region bounded by the given curves about the \(x\) -axis. In each case, sketch the region and a typical disk element. \(y=e^{x}, y=0, x=0, x=\ln 2\)

(a) Assume that \(a>0\). Evaluate \(\int_{0}^{a} x d x\), using the fact that the region bounded by \(y=x\) and the \(x\) -axis between 0 to \(a\) is a triangle. (b) Assume that \(a>0\). Evaluate \(\int_{0}^{a} x d x\) by approximating the region bounded by \(y=x\) and the \(x\) -axis from 0 to \(a\) with rectangles. Use equal subintervals and take right endpoints.

Compute the indefinite integrals. $$ \int\left(1+3 x^{2}\right) d x $$

Find the volumes of the solids obtained by rotating the region bounded by the given curves about the \(x\) -axis. In each case, sketch the region and a typical disk element. \(y=\sec x,-\frac{\pi}{3} \leq x \leq \frac{\pi}{3}, y=0\)

Assume that \(0

Compute the indefinite integrals. $$ \int\left(x^{3}-4\right) d x $$

Find the volumes of the solids obtained by rotating the region bounded by the given curves about the \(x\) -axis. In each case, sketch the region and a typical disk element. \(y=\sqrt{1-x^{2}}, 0 \leq x \leq 1, y=0\)

Compute the indefinite integrals. $$ \int\left(\frac{1}{3} x^{2}-\frac{1}{2} x\right) d x $$

In Problems 41-46, find the volumes of the solids obtained by rotating the region bounded by the given curves about the \(x\) -axis. In each case, sketch the region together with a typical disk element. \(y=x^{2}, y=x, 0 \leq x \leq 1\)

As definite integrals. Note that (1) \(P=\left[x_{0}, x_{1}, \ldots, x_{n}\right]\) is a partition of the indicated interval, (2) \(c_{k} \in\left[x_{k-1}, x_{k}\right]\), and (3) \(\Delta x_{k}=x_{k}-x_{k-1}\). \(\lim _{\|P\| \rightarrow 0} \sum_{k=1}^{n} 2 c_{k}^{3} \Delta x_{k}\), where \(P\) is a partition of \([1,2]\)

Compute the indefinite integrals. $$ \int\left(4 x^{3}+5 x^{2}\right) d x $$

Find the volumes of the solids obtained by rotating the region bounded by the given curves about the \(x\) -axis. In each case, sketch the region together with a typical disk element. \(y=2-x^{3}, y=2+x^{3}, 0 \leq x \leq 1\)

As definite integrals. Note that (1) \(P=\left[x_{0}, x_{1}, \ldots, x_{n}\right]\) is a partition of the indicated interval, (2) \(c_{k} \in\left[x_{k-1}, x_{k}\right]\), and (3) \(\Delta x_{k}=x_{k}-x_{k-1}\). \(\lim _{\|P\| \rightarrow 0} \sum_{k=1}^{n} \sqrt{c_{k}} \Delta x_{k}\), where \(P\) is a partition of \([1,4]\)

Compute the indefinite integrals. $$ \int\left(\frac{1}{2} x^{2}+3 x-\frac{1}{3}\right) d x $$

Find the volumes of the solids obtained by rotating the region bounded by the given curves about the \(x\) -axis. In each case, sketch the region together with a typical disk element. \(y=e^{x}, y=e^{-x}, 0 \leq x \leq 2\)

As definite integrals. Note that (1) \(P=\left[x_{0}, x_{1}, \ldots, x_{n}\right]\) is a partition of the indicated interval, (2) \(c_{k} \in\left[x_{k-1}, x_{k}\right]\), and (3) \(\Delta x_{k}=x_{k}-x_{k-1}\). \(\lim _{\mid P \| \rightarrow 0} \sum_{k=1}^{n}\left(2 c_{k}-1\right) \Delta x_{k}\), where \(P\) is a partition of \([-3,2]\)

Compute the indefinite integrals. $$ \int\left(\frac{1}{2} x^{5}+2 x^{3}-1\right) d x $$

Find the volumes of the solids obtained by rotating the region bounded by the given curves about the \(x\) -axis. In each case, sketch the region together with a typical disk element. \(y=\sqrt{1-x^{2}}, y=1, x=1\) (in the first quadrant)

As definite integrals. Note that (1) \(P=\left[x_{0}, x_{1}, \ldots, x_{n}\right]\) is a partition of the indicated interval, (2) \(c_{k} \in\left[x_{k-1}, x_{k}\right]\), and (3) \(\Delta x_{k}=x_{k}-x_{k-1}\). \(\lim _{\|P\| \rightarrow 0} \sum_{k=1}^{n} \frac{1}{c_{k}+1} \Delta x_{k}\), where \(P\) is a partition of \([1,2]\)

Compute the indefinite integrals. $$ \int \frac{2 x^{2}-x}{\sqrt{x}} d x $$

Compute the indefinite integrals. $$ \int \frac{x^{3}+3 x}{2 \sqrt{x}} d x $$

Find the volumes of the solids obtained by rotating the region bounded by the given curves about the \(x\) -axis. In each case, sketch the region together with a typical disk element. \(y=\frac{1}{x}, x=0, y=1, y=2\) (in the first quadrant)

As definite integrals. Note that (1) \(P=\left[x_{0}, x_{1}, \ldots, x_{n}\right]\) is a partition of the indicated interval, (2) \(c_{k} \in\left[x_{k-1}, x_{k}\right]\), and (3) \(\Delta x_{k}=x_{k}-x_{k-1}\). \(\lim _{\|P\| \rightarrow 0} \sum_{k=1}^{n}\left(\sin c_{k}\right) \Delta x_{k}\), where \(P\) is a partition of \([0, \pi]\)

Compute the indefinite integrals. $$ \int x^{2} \sqrt{x} d x $$

In Problems 47-52, find the volumes of the solids obtained by rotating the region bounded by the given curves about the \(y\) -axis. In each case, sketch the region together with a typical disk element. \(y=\sqrt{x}, y=2, x=0\)

As definite integrals. Note that (1) \(P=\left[x_{0}, x_{1}, \ldots, x_{n}\right]\) is a partition of the indicated interval, (2) \(c_{k} \in\left[x_{k-1}, x_{k}\right]\), and (3) \(\Delta x_{k}=x_{k}-x_{k-1}\). \(\lim _{\|P\| \rightarrow 0} \sum_{k=1}^{n} e^{c_{k}} \Delta x_{k}\), where \(P\) is a partition of \([-5,2]\)

Compute the indefinite integrals. $$ \int\left(1+x^{3}\right) \sqrt{x} d x $$

Find the volumes of the solids obtained by rotating the region bounded by the given curves about the \(y\) -axis. In each case, sketch the region together with a typical disk element. \(y=x^{2}, y=4, x=0\) (in the first quadrant)

Express the definite integrals as limits of Riemann sums. $$ \int_{-2}^{-1} \frac{x^{2}}{1+x^{2}} d x $$

Compute the indefinite integrals. $$ \int\left(x^{7 / 2}+x^{2 / 7}\right) d x $$

Find the volumes of the solids obtained by rotating the region bounded by the given curves about the \(y\) -axis. In each case, sketch the region together with a typical disk element. \(y=\ln (x+1), y=\ln 3, x=0\)

Express the definite integrals as limits of Riemann sums. $$ \int_{2}^{6}(x+1)^{1 / 3} d x $$

Compute the indefinite integrals. $$ \int\left(x^{3 / 5}+x^{5 / 3}\right) d x $$

Find the volumes of the solids obtained by rotating the region bounded by the given curves about the \(y\) -axis. In each case, sketch the region together with a typical disk element. \(y=\sqrt{x}, y=x, 0 \leq x \leq 1\)

Express the definite integrals as limits of Riemann sums. $$ \int_{1}^{3} e^{-2 x} d x $$

Compute the indefinite integrals. $$ \int\left(\sqrt{x}+\frac{1}{\sqrt{x}}\right) d x $$

Find the volumes of the solids obtained by rotating the region bounded by the given curves about the \(y\) -axis. In each case, sketch the region together with a typical disk element. \(y=x^{2}, y=\sqrt{x}, 0 \leq x \leq 1\)

Express the definite integrals as limits of Riemann sums. $$ \int_{1}^{e} \ln x d x $$

Compute the indefinite integrals. $$ \int\left(3 x^{1 / 3}+\frac{1}{3 x^{1 / 3}}\right) d x $$

Find the volumes of the solids obtained by rotating the region bounded by the given curves about the \(y\) -axis. In each case, sketch the region together with a typical disk element. \(y=\frac{1}{x}, x=0, y=\frac{1}{2}, y=1\)

Express the definite integrals as limits of Riemann sums. $$ \int_{0}^{\pi} \cos \frac{2 x}{\pi} d x $$

Compute the indefinite integrals. $$ \int(x-1)(x+1) d x $$

Express the definite integrals as limits of Riemann sums. \(\int_{0}^{5} g(x) d x\), where \(g(x)\) is a continuous function on \([0,5]\)

Compute the indefinite integrals. $$ \int(x-1)^{2} d x $$