Chapter 5

Use the Substitution Formula in Theorem 6 to evaluate the integrals in Exercises \(1-24 .\) $$ \text { a. }\int_{0}^{3} \sqrt{y+1} d y \quad \text { b. } \int_{-1}^{0} \sqrt{y+1} d y $$

Evaluate the indefinite integrals in Exercises \(1-12\) by using the given substitutions to reduce the integrals to standard form. $$ \int \sin 3 x d x, \quad u=3 x $$

Write the sums in Exercises \(1-6\) without sigma notation. Then evaluate them. $$ \sum_{k=1}^{2} \frac{6 k}{k+1} $$

Evaluate the integrals in Exercises \(1-26\) $$ \int_{-2}^{0}(2 x+5) d x $$

In Exercises \(1-4\) use finite approximations to estimate the area under the graph of the function using a. a lower sum with two rectangles of equal width. b. a lower sum with four rectangles of equal width. c. an upper sum with two rectangles of equal width. d. an upper sum with four rectangles of equal width. $$ f(x)=x^{2} \text { between } x=0 \text { and } x=1 $$

Use the Substitution Formula in Theorem 6 to evaluate the integrals in Exercises \(1-24 .\) $$ \text { a. }\int_{0}^{1} r \sqrt{1-r^{2}} d r \quad \text { b. } \int_{-1}^{1} r \sqrt{1-r^{2}} d r $$

Evaluate the indefinite integrals in Exercises \(1-12\) by using the given substitutions to reduce the integrals to standard form. $$ \int x \sin \left(2 x^{2}\right) d x, \quad u=2 x^{2} $$

Write the sums in Exercises \(1-6\) without sigma notation. Then evaluate them. $$ \sum_{k=1}^{3} \frac{k-1}{k} $$

Evaluate the integrals in Exercises \(1-26\) $$ \int_{-3}^{4}\left(5-\frac{x}{2}\right) d x $$

Express the limits in Exercises \(1-8\) as definite integrals. $$ \lim _{\|P\| \rightarrow 0} \sum_{k=1}^{n} 2 c_{k}^{3} \Delta x_{k}, \text { where } P \text { is a partition of }[-1,0] $$

In Exercises \(1-4\) use finite approximations to estimate the area under the graph of the function using a. a lower sum with two rectangles of equal width. b. a lower sum with four rectangles of equal width. c. an upper sum with two rectangles of equal width. d. an upper sum with four rectangles of equal width. $$ f(x)=x^{3} \text { between } x=0 \text { and } x=1 $$

Evaluate the indefinite integrals in Exercises \(1-12\) by using the given substitutions to reduce the integrals to standard form. $$ \int \sec 2 t \tan 2 t d t, \quad u=2 t $$

Write the sums in Exercises \(1-6\) without sigma notation. Then evaluate them. $$ \sum_{k=1}^{4} \cos k \pi $$

Evaluate the integrals in Exercises \(1-26\) $$ \int_{0}^{4}\left(3 x-\frac{x^{3}}{4}\right) d x $$

Evaluate the indefinite integrals in Exercises \(1-12\) by using the given substitutions to reduce the integrals to standard form. $$ \int\left(1-\cos \frac{t}{2}\right)^{2} \sin \frac{t}{2} d t, \quad u=1-\cos \frac{t}{2} $$

Write the sums in Exercises \(1-6\) without sigma notation. Then evaluate them. $$ \sum_{k=1}^{5} \sin k \pi $$

Evaluate the integrals in Exercises \(1-26\) $$ \int_{-2}^{2}\left(x^{3}-2 x+3\right) d x $$

Express the limits in Exercises \(1-8\) as definite integrals. $$ \lim _{\|P\| \rightarrow 0} \sum_{k=1}^{n}\left(\frac{1}{c_{k}}\right) \Delta x_{k}, \text { where } P \text { is a partition of }[1,4] $$

In Exercises \(1-4\) use finite approximations to estimate the area under the graph of the function using a. a lower sum with two rectangles of equal width. b. a lower sum with four rectangles of equal width. c. an upper sum with two rectangles of equal width. d. an upper sum with four rectangles of equal width. $$ f(x)=4-x^{2} \text { between } x=-2 \text { and } x=2 $$

Evaluate the indefinite integrals in Exercises \(1-12\) by using the given substitutions to reduce the integrals to standard form. $$ \int 28(7 x-2)^{-5} d x, \quad u=7 x-2 $$

Write the sums in Exercises \(1-6\) without sigma notation. Then evaluate them. $$ \sum_{k=1}^{3}(-1)^{k+1} \sin \frac{\pi}{k} $$

Evaluate the integrals in Exercises \(1-26\) $$ \int_{0}^{1}\left(x^{2}+\sqrt{x}\right) d x $$

Express the limits in Exercises \(1-8\) as definite integrals. $$ \lim _{\|P\| \rightarrow 0} \sum_{k=1}^{n} \frac{1}{1-c_{k}} \Delta x_{k}, \text { where } P \text { is a partition of }[2,3] $$

Using rectangles whose height is given by the value of the function at the midpoint of the rectangle's base the midpoint rule estimate the area under the graphs of the following functions, using first two and then four rectangles. $$ f(x)=x^{2} \text { between } x=0 \text { and } x=1 $$

Use the Substitution Formula in Theorem 6 to evaluate the integrals in Exercises \(1-24 .\) $$ \text { a. }\int_{0}^{\sqrt{7}} t\left(t^{2}+1\right)^{1 / 3} d t \quad \text { b. } \int_{-\sqrt{7}}^{0} t\left(t^{2}+1\right)^{1 / 3} d t $$

Evaluate the indefinite integrals in Exercises \(1-12\) by using the given substitutions to reduce the integrals to standard form. $$ \int x^{3}\left(x^{4}-1\right)^{2} d x, \quad u=x^{4}-1 $$

Write the sums in Exercises \(1-6\) without sigma notation. Then evaluate them. $$ \sum_{k=1}^{4}(-1)^{k} \cos k \pi $$

Evaluate the integrals in Exercises \(1-26\) $$ \int_{0}^{5} x^{3 / 2} d x $$

Express the limits in Exercises \(1-8\) as definite integrals. $$ \lim _{\|P\| \rightarrow 0} \sum_{k=1}^{n} \sqrt{4-c_{k}^{2}} \Delta x_{k}, \text { where } P \text { is a partition of }[0,1] $$

Using rectangles whose height is given by the value of the function at the midpoint of the rectangle's base the midpoint rule estimate the area under the graphs of the following functions, using first two and then four rectangles. $$ f(x)=x^{3} \text { between } x=0 \text { and } x=1 $$

Evaluate the indefinite integrals in Exercises \(1-12\) by using the given substitutions to reduce the integrals to standard form. $$ \int \frac{9 r^{2} d r}{\sqrt{1-r^{3}}}, \quad u=1-r^{3} $$

Which of the following express \(1+2+4+8+16+32\) in sigma notation? $$ \text { a. }\sum_{k=1}^{6} 2^{k-1} \quad \text { b. } \sum_{k=0}^{5} 2^{k} \quad \text { c. } \sum_{k=1}^{4} 2^{k+1} $$

Evaluate the integrals in Exercises \(1-26\) $$ \int_{1}^{32} x^{-6 / 5} d x $$

Express the limits in Exercises \(1-8\) as definite integrals. $$ \lim _{\|P\| \rightarrow 0} \sum_{k=1}^{n}\left(\sec c_{k}\right) \Delta x_{k}, \text { where } P \text { is a partition of }[-\pi / 4,0] $$

Using rectangles whose height is given by the value of the function at the midpoint of the rectangle's base the midpoint rule estimate the area under the graphs of the following functions, using first two and then four rectangles. $$ f(x)=1 / x \text { between } x=1 \text { and } x=5 $$

Evaluate the indefinite integrals in Exercises \(1-12\) by using the given substitutions to reduce the integrals to standard form. $$ \int 12\left(y^{4}+4 y^{2}+1\right)^{2}\left(y^{3}+2 y\right) d y, \quad u=y^{4}+4 y^{2}+1 $$

Which of the following express \(1-2+4-8+16-32\) in sigma notation? $$ \text { a. }\sum_{k=1}^{6}(-2)^{k-1} \quad \text { b. } \sum_{k=0}^{5}(-1)^{k} 2^{k} \quad \text { c. } \sum_{k=-2}^{3}(-1)^{k+1} 2^{k+2} $$

Evaluate the integrals in Exercises \(1-26\) $$ \int_{-2}^{-1} \frac{2}{x^{2}} d x $$

Express the limits in Exercises \(1-8\) as definite integrals. $$ \lim _{\|P\| \rightarrow 0} \sum_{k=1}^{n}\left(\tan c_{k}\right) \Delta x_{k}, \text { where } P \text { is a partition of }[0, \pi / 4] $$

Using rectangles whose height is given by the value of the function at the midpoint of the rectangle's base the midpoint rule estimate the area under the graphs of the following functions, using first two and then four rectangles. $$ f(x)=4-x^{2} \text { between } x=-2 \text { and } x=2 $$

Evaluate the indefinite integrals in Exercises \(1-12\) by using the given substitutions to reduce the integrals to standard form. $$ \int \sqrt{x} \sin ^{2}\left(x^{3 / 2}-1\right) d x, \quad u=x^{3 / 2}-1 $$

Which formula is not equivalent to the other two? $$ \text { a. }\sum_{k=2}^{4} \frac{(-1)^{k-1}}{k-1} \quad \text { b. } \sum_{k=0}^{2} \frac{(-1)^{k}}{k+1} \quad \text { c. } \sum_{k=1}^{1} \frac{(-1)^{k}}{k+2} $$

Evaluate the integrals in Exercises \(1-26\) $$ \int_{0}^{\pi} \sin x d x $$

Suppose that \(f\) and \(g\) are integrable and that $$\int_{1}^{2} f(x) d x=-4, \int_{1}^{5} f(x) d x=6, \int_{1}^{5} g(x) d x=8$$ Use the rules in Table 5.3 to find $$ \begin{array}{ll}{\text { a. } \int_{2}^{2} g(x) d x} & {\text { b. } \int_{5}^{1} g(x) d x} \\ {\text { c. } \int_{1}^{2} 3 f(x) d x} & {\text { d. } \int_{2}^{5} f(x) d x} \\ {\text { e. } \int_{1}^{5}[f(x)-g(x)] d x} & {\text { f. } \int_{1}^{5}[4 f(x)-g(x)] d x}\end{array} $$

Evaluate the indefinite integrals in Exercises \(1-12\) by using the given substitutions to reduce the integrals to standard form. $$ \int \frac{1}{x^{2}} \cos ^{2}\left(\frac{1}{x}\right) d x, \quad u=-\frac{1}{x} $$

Evaluate the integrals in Exercises \(1-26\) $$ \int_{0}^{\pi}(1+\cos x) d x $$

Use the Substitution Formula in Theorem 6 to evaluate the integrals in Exercises \(1-24 .\) $$ \mathbf{a}\int_{0}^{\pi / 6}(1-\cos 3 t) \sin 3 t d t \quad \mathbf{b} . \int_{\pi / 6}^{\pi / 3}(1-\cos 3 t) \sin 3 t d t $$

Evaluate the indefinite integrals in Exercises \(1-12\) by using the given substitutions to reduce the integrals to standard form. $$ \begin{array}{l}{\int \csc ^{2} 2 \theta \cot 2 \theta d \theta} \\ {\text { a. Using } u=\cot 2 \theta \quad \text { b. Using } u=\csc 2 \theta}\end{array} $$

Express the sums in Exercises \(11-16\) in sigma notation. The form of your answer will depend on your choice of the lower limit of summation. $$ 1+2+3+4+5+6 $$

Evaluate the integrals in Exercises \(1-26\) $$ \int_{0}^{\pi / 3} 2 \sec ^{2} x d x $$