Chapter 5

Use the Substitution Formula in Theorem 7 to evaluate the integrals in Exercises \(1-24 .\) $$ \int_{0}^{3} \sqrt{y+1} d y $$

Evaluate the integrals in Exercises \(1-28\). $$\int_{0}^{2} x(x-3) d x$$

Evaluate the indefinite integrals in Exercises \(1-16\) by using the given substitutions to reduce the integrals to standard form. $$ \int 2(2 x+4)^{5} d x, \quad u=2 x+4 $$

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

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

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 \(1-28\). $$\int_{-1}^{1}\left(x^{2}-2 x+3\right) d x$$

Evaluate the indefinite integrals in Exercises \(1-16\) by using the given substitutions to reduce the integrals to standard form. $$ \int 7 \sqrt{7 x-1} d x, u=7 x-1 $$

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 $$\begin{array}{l}{\text { a. a lower sum with two rectangles of equal width. }} \\ {\text { b. a lower sum with four rectangles of equal width. }} \\\ {\text { c. an upper sum with two rectangles of equal width. }} \\ {\text { d. an upper sum with four rectangles of equal width. }}\end{array}$$ \(f(x)=x^{3}\) between \(x=0\) and \(x=1\)

Evaluate the integrals in Exercises \(1-28\). $$\int_{-2}^{2} \frac{3}{(x+3)^{4}} d x$$

Evaluate the indefinite integrals in Exercises \(1-16\) by using the given substitutions to reduce the integrals to standard form. $$ \int 2 x\left(x^{2}+5\right)^{-4} d x, \quad u=x^{2}+5 $$

Express the limits in Exercises \(1-8\) as definite integrals. $$ \lim _{|P| \rightarrow 0} \sum_{k=1}^{n}\left(c_{k}^{2}-3 c_{k}\right) \Delta x_{k}, \text { where } P \text { is a partition of }[-7,5] $$

In Exercises \(1-4,\) use finite approximations to estimate the area under the graph of the function using $$\begin{array}{l}{\text { a. a lower sum with two rectangles of equal width. }} \\ {\text { b. a lower sum with four rectangles of equal width. }} \\\ {\text { c. an upper sum with two rectangles of equal width. }} \\ {\text { d. an upper sum with four rectangles of equal width. }}\end{array}$$ \(f(x)=1 / x\) between \(x=1\) and \(x=5\)

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

Use the Substitution Formula in Theorem 7 to evaluate the integrals in Exercises \(1-24 .\) $$ \int_{0}^{\pi} 3 \cos ^{2} x \sin x d x \quad \text { b. } \int_{2 \pi}^{3 \pi} 3 \cos ^{2} x \sin x d x $$

Evaluate the integrals in Exercises \(1-28\). $$\int_{-1}^{1} x^{299} d x$$

Evaluate the indefinite integrals in Exercises \(1-16\) by using the given substitutions to reduce the integrals to standard form. $$ \int \frac{4 x^{3}}{\left(x^{4}+1\right)^{2}} d x, \quad u=x^{4}+1 $$

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 $$\begin{array}{l}{\text { a. a lower sum with two rectangles of equal width. }} \\ {\text { b. a lower sum with four rectangles of equal width. }} \\\ {\text { c. an upper sum with two rectangles of equal width. }} \\ {\text { d. an upper sum with four rectangles of equal width. }}\end{array}$$ \(f(x)=4-x^{2}\) between \(x=-2\) and \(x=2\)

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

Use the Substitution Formula in Theorem 7 to evaluate the integrals in Exercises \(1-24 .\) $$ \int_{-1}^{1} \frac{5 r}{\left(4+r^{2}\right)^{2}} d r \quad \text { b. } \int_{0}^{1} \frac{5 r}{\left(4+r^{2}\right)^{2}} d r $$

Evaluate the integrals in Exercises \(1-28\). $$\int_{1}^{4}\left(3 x^{2}-\frac{x^{3}}{4}\right) d x$$

Evaluate the indefinite integrals in Exercises \(1-16\) by using the given substitutions to reduce the integrals to standard form. $$ \int(3 x+2)\left(3 x^{2}+4 x\right)^{4} d x, u=3 x^{2}+4 x $$

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 each of 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. \(f(x)=x^{2}\) between \(x=0\) and \(x=1\)

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

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

Evaluate the integrals in Exercises \(1-28\). $$\int_{-2}^{3}\left(x^{3}-2 x+3\right) d x$$

Evaluate the indefinite integrals in Exercises \(1-16\) by using the given substitutions to reduce the integrals to standard form. $$ \int \frac{(1+\sqrt{x})^{1 / 3}}{\sqrt{x}} d x, u=1+\sqrt{x} $$

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 each of 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. \(f(x)=x^{3}\) between \(x=0\) and \(x=1\)

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 \(1-28\). $$\int_{0}^{1}\left(x^{2}+\sqrt{x}\right) d x$$

Evaluate the indefinite integrals in Exercises \(1-16\) by using the given substitutions to reduce the integrals to standard form. $$ \int \sin 3 x d x, u=3 x $$

Using rectangles each of 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. \(f(x)=1 / x\) between \(x=1\) and \(x=5\)

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

Evaluate the integrals in Exercises \(1-28\). $$\int_{1}^{32} x^{-6 / 5} d x$$

Evaluate the indefinite integrals in Exercises \(1-16\) by using the given substitutions to reduce the integrals to standard form. $$ \int x \sin \left(2 x^{2}\right) d x, \quad u=2 x^{2} $$

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 each of 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. \(f(x)=4-x^{2}\) between \(x=-2\) and \(x=2\)

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

Evaluate the integrals in Exercises \(1-28\). $$\int_{0}^{\pi / 3} 2 \sec ^{2} x d x$$

Evaluate the indefinite integrals in Exercises \(1-16\) by using the given substitutions to reduce the integrals to standard form. $$ \int \sec 2 t \tan 2 t d t, \quad u=2 t $$

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

Use the Substitution Formula in Theorem 7 to evaluate the integrals in Exercises \(1-24 .\) $$ { a. }\int_{0}^{1} \frac{x^{3}}{\sqrt{x^{4}+9}} d x \quad \text { b. } \int_{-1}^{0} \frac{x^{3}}{\sqrt{x^{4}+9}} d x$$

Evaluate the integrals in Exercises \(1-28\). $$\int_{0}^{\pi}(1+\cos x) d x$$

Evaluate the indefinite integrals in Exercises \(1-16\) by using the given substitutions to reduce the integrals to standard form. $$ \int\left(1-\cos \frac{t}{2}\right)^{2} \sin \frac{t}{2} d t, \quad u=1-\cos \frac{t}{2} $$

Which formula is not equivalent to the other two? $$ a. \sum_{k=1}^{4}(k-1)^{2} \quad \text { b. } \sum_{k=-1}^{3}(k+1)^{2} \quad \text { c. } \sum_{k=-3}^{-1} k^{2}$$

Evaluate the integrals in Exercises \(1-28\). $$\int_{\pi / 4}^{3 \pi / 4} \csc \theta \cot \theta d \theta$$