Chapter 6

Find the length of the straight line $$ y=m x $$ from \(x=0\) to \(x=a\), where \(m\) and \(a\) are positive constants, by each of the following methods: (a) planar geometry (b) the integral formula for the lengths of curves, derived in Subsection \(6.3 .5\)

Use a graph to interpret the definite integral in terms of areas. Do not compute the integrals. $$ \int_{0}^{3}(2 x+1) d x $$

Compute the indefinite integrals. $$ \int(x-2)(3-x) d x $$

Find the length of the curve $$ y^{2}=x^{3} $$ from \(x=1\) to \(x=4\)

Use a graph to interpret the definite integral in terms of areas. Do not compute the integrals. $$ \int_{-1}^{2}\left(x^{2}-1\right) d x $$

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

Find the length of the curve $$ 2 y^{2}=3 x^{3} $$ from \(x=0\) to \(x=1\).

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

Find the length of the curve $$ y=\frac{x^{3}}{6}+\frac{1}{2 x} $$ from \(x=1\) to \(x=3\).

Use a graph to interpret the definite integral in terms of areas. Do not compute the integrals. $$ \int_{0}^{5} e^{-x} d x $$

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

Find the length of the curve $$ y=\frac{x^{4}}{4}+\frac{1}{8 x^{2}} $$ from \(x=2\) to \(x=4\)

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

In Problems 59-62, set up, but do not evaluate, the integrals for the lengths of the following curves: \(y=x^{2},-1 \leq x \leq 1\)

Use a graph to interpret the definite integral in terms of areas. Do not compute the integrals. $$ \int_{1 / 2}^{4} \ln x d x $$

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

Set up, but do not evaluate, the integrals for the lengths of the following curves: \(y=\sin x, 0 \leq x \leq \frac{\pi}{2}\)

Use a graph to interpret the definite integral in terms of areas. Do not compute the integrals. $$ \int_{-3}^{2}\left(1-\frac{1}{2} x\right) d x $$

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

Set up, but do not evaluate, the integrals for the lengths of the following curves: \(y=e^{-x}, 0 \leq x \leq 1\)

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. $$ \int_{-2}^{5}|x| d x $$

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

Set up, but do not evaluate, the integrals for the lengths of the following curves: \(y=\ln x, 1 \leq x \leq e\)

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. $$ \int_{-3}^{3} \sqrt{9-x^{2}} d x $$

Compute the indefinite integrals. $$ \int \sin (2 x) d x $$

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. $$ \int_{2}^{5}\left(\frac{1}{2} x-4\right) d x $$

Compute the indefinite integrals. $$ \int \sin \frac{1-x}{3} d x $$

A cable that hangs between two poles at \(x=-M\) and \(x=M\) takes the shape of a catenary, with equation $$ y=\frac{1}{2 a}\left(e^{a x}+e^{-a x}\right) $$ where \(a\) is a positive constant. Compute the length of the cable when \(a=1\) and \(M=\ln 2\).

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. $$ \int_{1 / 2}^{1} \sqrt{1-x^{2}} d x $$

Compute the indefinite integrals. $$ \int \cos (3 x) d x $$

Show that if $$ f(x)=\frac{e^{x}+e^{-x}}{2} $$ then the length of the curve \(f(x)\) between \(x=0\) and \(x=a\) for any \(a>0\) is given by \(f^{\prime}(a)\).

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

Compute the indefinite integrals. $$ \int \cos \frac{2-4 x}{5} d x $$

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. $$ \int_{0}^{1} \sqrt{2-x^{2}} d x $$

Compute the indefinite integrals. $$ \int \sec ^{2}(3 x) d x $$

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. $$ \int_{-3}^{0}\left(4-\sqrt{9-x^{2}}\right) d x $$

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

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

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

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

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

Find \(\int_{-3}^{-3} e^{-x^{2} / 2} d x\).

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

Find \(\int_{-2}^{2} \frac{x^{3}}{3} d x\)

Compute the indefinite integrals. $$ \int \cot (3 x) d x $$

Find \(\int_{-5}^{5} 2 x^{5} d x\).

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

Compute the indefinite integrals. $$ \int\left(\cot x-\csc ^{2} x\right) d x $$

Explain geometrically why $$\int_{1}^{2} x^{2} d x=\int_{0}^{2} x^{2} d x-\int_{0}^{1} x^{2} d x$$ and show that (6.4) can be written as $$\int_{1}^{2} x^{2} d x=\int_{1}^{0} x^{2} d x+\int_{0}^{2} x^{2} d x$$ Relate (6.5) to addition property (5).

Compute the indefinite integrals. $$ \int \frac{4}{1+x^{2}} d x $$