Chapter 6

In Problems , use an area formula from geometry to find the value of each integral by interpreting it as the (signed) area under the graph of an appropriately chosen function. $$ \int_{1 / 2}^{1} \sqrt{1-x^{2}} d x $$

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. \(y=4-x^{2}, y=0, x=0\) (in the first quadrant)

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

$$ \int_{-2}^{2}\left(\sqrt{4-x^{2}}-2\right) d x $$

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. \(y=\sqrt{x}, y=0, x=1\)

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 and a typical disk element. \(y=x, 0 \leq x \leq 1\)

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 and a typical disk element. \(y=e^{x}, y=0, x=0, x=\ln 2\)

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 and a typical disk element. \(y=x^{2},-1 \leq x \leq 1\)

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

In Problems , use an area formula from geometry to find the value of each integral by interpreting it as the (signed) area under the graph of an appropriately chosen function. $$ \int_{-1}^{2}(2-|x|) d x $$

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 and a typical disk element. \(y=\sqrt{1-x^{2}}, 0 \leq x \leq 1, y=0\)

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

In Problems , use an area formula from geometry to find the value of each integral by interpreting it as the (signed) area under the graph of an appropriately chosen function. $$ \int_{0}^{2}|x-1| d x $$

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. \(y=x^{2}, y=x, 0 \leq x \leq 1\)

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

In Problems , use an area formula from geometry to find the value of each integral by interpreting it as the (signed) area under the graph of an appropriately chosen function. $$ \int_{0}^{2}(2+x) d x $$

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. \(y=2-x^{3}, y=2+x^{3}, 0 \leq x \leq 1\)

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

Use the definition of the Riemann integral in terms of Riemann sums to prove property (3) of definite integrals. That is, if \(f(x)\) is continuous on \([a, b]\) and \(k\) is any constant, then: $$ \int^{b} k f(x) d x=k \int^{b} f(x) d x $$

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. \(y=e^{x}, y=e^{-x}, 0 \leq x \leq 2\)

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

Use a diagram to explain why, if \(f(x)\) is continuous on an interval that contains all of the points \(a, b, c\), then $$ \int_{a}^{b} f(x) d x=\int_{a}^{c} f(x) d x+\int_{c}^{b} f(x) d x $$ That is, derive property (5) of definite integrals. You should consider the cases (a) \(b

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. \(y=\sqrt{1-x^{2}}, y=1,-1 \leq x \leq 1\)

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

Given that \(\int_{0}^{u} x^{2} d x=\frac{1}{3} a^{3}\) evaluate the following: (a) \(\int_{0}^{1} \frac{1}{2} x^{2} d x\) (b) \(\int_{0}^{-1} 3 x^{2} d x\) (c) \(\int_{-1}^{2} \frac{1}{3} x^{2} d x\) (d) \(\int_{1}^{1} 3 x^{2} d x\) (e) \(\int_{-2}^{3}(x+1)^{2} d x\) (f) \(\int_{2}^{4}(x-2)^{2} d x\)

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. \(y=x, y=1,0 \leq x \leq 1\)

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

$$ \text { Find } \int_{2}^{2} \cos \left(3 x^{2}\right) d x $$

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 together with a typical disk element. \(y=1-x^{2}, y=1,-1 \leq x \leq 1\)

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

$$ \text { Find } \int_{-3}^{-3} e^{-x^{2} / 2} d x \text { . } $$

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. \(y=\sqrt{x}, y=2, x=0\)

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

$$ \text { Find } \int_{-1}^{1} 3 x d x \text { . } $$

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. \(y=x^{2}, y=4, x=0\) (in the first quadrant)

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

$$ \text { Find } \int_{-1}^{1} 3 x^{5} d x $$

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. \(y=|x|, y=1,-1 \leq x \leq 1\)

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

$$ \text { Find } \int_{0}^{2}(x-1)^{3} d x \text { . } $$

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. \(y=\sqrt{x}, y=x, 0 \leq x \leq 1\)

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

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. \(y=x^{3}, y=x^{2}, 0 \leq x \leq 1\)

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

Given that \(\int_{0}^{a} x^{3} d x=\frac{1}{4} a^{4}\), evaluate the following integrals: (a) \(\int_{0}^{2} x^{3} d x\) (b) \(\int_{0}^{1} 2 x^{3} d x\) (c) \(\int_{-1}^{1} 2 x^{3} d x\) (d) \(\int_{-1}^{1}(x+1)^{3} d x\) (e) \(\int_{1}^{2} 2(x+2)^{3} d x\).

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. \(y=|x|, y=0,-1 \leq x \leq 1\)

Compute the indefinite integrals. $$ \int 2 e^{3 x} d x $$