Chapter 4

Calculate total cost (disregarding any fixed costs) or total profit. Total profit from marginal profit. A concert promoter sells \(x\) tickets and has a marginal-profit function given by $$P^{\prime}(x)=2 x-150$$ where \(P^{\prime}(x)\) is in dollars per ticket. This means that the rate of change of total profit with respect to the number of tickets sold, \(x\), is \(P^{\prime}(x)\). Find the total profit from the sale of the first 300 tickets.

Evaluate. (Be sure to check by differentiating!) $$ \int\left(8+x^{3}\right)^{5} 3 x^{2} d x $$

Find each antiderivative using Table 1. $$ \int x e^{-3 x} d x $$

Evaluate using integration by parts or substitution. (Assume \(u>0\) in \(\ln\) u. Check by differentiating. $$ \int 4 x e^{4 x} d x $$

Find each integral. $$ \int x^{6} d x $$

Find the area under the given curve over the indicated interval. $$ y=4 ; \quad[1,3] $$

Find the area under the graph of \(f\) over [1,5]. $$ f(x)=\left\\{\begin{array}{lll} 2 x+1, & \text { for } & x \leq 3 \\ 10-x, & \text { for } & x>3 \end{array}\right. $$

Calculate total cost (disregarding any fixed costs) or total profit. Poyse Inc. has a marginal-profit function given by $$P^{\prime}(x)=-2 x+80$$ where \(P^{\prime}(x)\) is in dollars per unit. This means that the rate of change of total profit with respect to the number of units produced, \(x\), is \(P^{\prime}(x)\). Find the total profit from the production and sale of the first 40 units.

Evaluate. (Be sure to check by differentiating!) $$ \int\left(x^{2}-7\right)^{6} 2 x d x $$

Evaluate using integration by parts or substitution. (Assume \(u>0\) in \(\ln\) u. Check by differentiating. $$ \int 3 x e^{3 x} d x $$

Find each integral. $$ \int x^{7} d x $$

Find the area under the given curve over the indicated interval. $$ y=5 ; \quad[1,3] $$

Find the area under the graph of \(f\) over [1,5]. $$ f(x)=\left\\{\begin{array}{ll} x+5, & \text { for } \quad x \leq 4 \\ 11-\frac{1}{2} x, & \text { for } \quad x>4 \end{array}\right. $$

Calculate total cost (disregarding any fixed costs) or total profit. Sylvie's Old World Cheeses has found that its marginal cost, in dollars per kilogram is $$C^{\prime}(x)=-0.003 x+4.25, \quad \text { for } x \leq 500$$ where \(x\) is the number of kilograms of cheese produced. Find the total cost of producing \(400 \mathrm{~kg}\) of cheese.

Evaluate. (Be sure to check by differentiating!) $$ \int\left(x^{2}-6\right)^{7} x d x $$

Find each integral. $$ \int 2 d x $$

Find the area under the given curve over the indicated interval. $$ y=2 x ; \quad[1,3] $$

Find the area under the graph of \(g\) over [-2,3] . $$ g(x)=\left\\{\begin{array}{ll} x^{2}+4, & \text { for } \quad x \leq 0 \\ 4-x, & \text { for } \quad x>0 \end{array}\right. $$

Calculate total cost (disregarding any fixed costs) or total profit. Redline Roasting has found that its marginal cost, in dollars per pound, is $$C^{\prime}(x)=-0.012 x+6.50, \quad \text { for } x \leq 300$$ where \(x\) is the number of pounds of coffee roasted. Find the total cost of roasting \(200 \mathrm{lb}\) of coffee.

Evaluate. (Be sure to check by differentiating!) $$ \int\left(x^{3}+1\right)^{4} x^{2} d x $$

Evaluate using integration by parts or substitution. (Assume \(u>0\) in \(\ln\) u. Check by differentiating. $$ \int x^{2}(2 x) d x $$

Find each integral. $$ \int 4 d x $$

Find the area under the given curve over the indicated interval. $$ y=x^{2} ; \quad[0,3] $$

Find the area under the graph of \(g\) over [-2,3] . $$ g(x)=\left\\{\begin{array}{lll} -x^{2}+5, & \text { for } & x \leq 0 \\ x+5, & \text { for } & x>0 \end{array}\right. $$

Calculate total cost (disregarding any fixed costs) or total profit. Cleo's Custom Fabrics has found that its marginal cost, in dollars per yard, is $$C^{\prime}(x)=-0.007 x+12, \quad \text { for } x \leq 350$$ where \(x\) is the number of yards of fabric produced. Find the total cost of producing \(200 \mathrm{yd}\) of this fabric.

Evaluate. (Be sure to check by differentiating!) $$ \int\left(3 t^{4}+2\right) t^{3} d t $$

Evaluate using integration by parts or substitution. (Assume \(u>0\) in \(\ln\) u. Check by differentiating. $$ \int x e^{5 x} d x $$

Find each integral. $$ \int x^{1 / 4} d x $$

Find the area under the given curve over the indicated interval. $$ y=x^{2} ; \quad[0,5] $$

Find the area under the graph of \(f\) over [-6,4] . $$ f(x)=\left\\{\begin{array}{lll} -x^{2}-6 x+7, & \text { for } & x<1 \\ \frac{3}{2} x-1, & \text { for } & x \geq 1 \end{array}\right. $$

Evaluate. (Be sure to check by differentiating!) $$ \int\left(2 t^{5}-3\right) t^{4} d t $$

Evaluate using integration by parts or substitution. (Assume \(u>0\) in \(\ln\) u. Check by differentiating. $$ \int 2 x e^{4 x} d x $$

Find each integral. $$ \int x^{1 / 3} d x $$

Find the area under the given curve over the indicated interval. $$ y=x^{3} ; \quad[0,2] $$

Find the area under the graph of \(f\) over [-6,4] . $$ f(x)=\left\\{\begin{array}{ll} -x-1, & \text { for } \quad x<-1 \\ -x^{2}+4 x+5, & \text { for } \quad x \geq-1 \end{array}\right. $$

Evaluate. (Be sure to check by differentiating!) $$ \int \frac{2}{1+2 x} d x $$

Evaluate using integration by parts or substitution. (Assume \(u>0\) in \(\ln\) u. Check by differentiating. $$ \int x e^{-2 x} d x $$

Find each integral. $$ \int\left(x^{2}+x-1\right) d x $$

Find the area under the given curve over the indicated interval. $$ y=x^{3} ; \quad[0,1] $$

Find the area represented by each definite integral. $$ \int_{0}^{4}|x-3| d x $$

Evaluate. (Be sure to check by differentiating!) $$ \int \frac{5}{5 x+7} d x $$

Evaluate using integration by parts or substitution. (Assume \(u>0\) in \(\ln\) u. Check by differentiating. $$ \int x e^{-x} d x $$

Find each integral. $$ \int\left(x^{2}-x+2\right) d x $$

Find the area under the given curve over the indicated interval. $$ y=1-x^{2} ; \quad[-1,1] $$

Find the area represented by each definite integral. $$ \int_{-1}^{1}|3 x-2| d x $$

Evaluate. (Be sure to check by differentiating!) $$ \int(\ln x)^{3} \frac{1}{x} d x, x>0 $$

Evaluate using integration by parts or substitution. (Assume \(u>0\) in \(\ln\) u. Check by differentiating. $$ \int x^{2} \ln x d x $$

Find each integral. $$ \int\left(2 t^{2}+5 t-3\right) d t $$

Find the area under the given curve over the indicated interval. $$ y=4-x^{2} ; \quad[-2,2] $$

Find the area represented by each definite integral. $$ \int_{0}^{2}\left|x^{3}-1\right| d x $$