91Ó°ÊÓ

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

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.

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

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

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

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

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.

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

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

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