91Ó°ÊÓ

Determine the following: $$\int 4 x^{3} d x$$

Determine the following: $$\int 7 d x$$

Combine the integrals into one integral, then evaluate the integral. $$\int_{.5}^{1.5}\left(-2 x-\frac{x^{3}}{3}\right) d x+2 \int_{.5}^{1.5}(\sqrt{x}+x) d x$$

A savings account pays \(4.25 \%\) interest compounded continuously. At what rate per year must money be deposited steadily in the account to accumulate a balance of $$\$ 100,000$$ after 10 years?

Volume of Solids of Revolution Find the volume of the solid of revolution generated by revolving about the \(x\) -axis the region under each of the following curves. \(y=\sqrt{r^{2}-x^{2}}\) from \(x=-r\) to \(x=r\) (generates a sphere of radius \(r\) )

Volume of Solids of Revolution Find the volume of the solid of revolution generated by revolving about the \(x\) -axis the region under each of the following curves. \(y=\sqrt{x}\) from \(x=0\) to \(x=4\) (The solid generated is called a paraboloid.)

A company's marginal cost function is \(.1 x^{2}-x+12\) dollars, where \(x\) denotes the number of units produced in 1 day. (a) Determine the increase in cost if the production level is raised from \(x=1\) to \(x=3\) units. (b) If \(C(1)=15\), determine \(C(3)\) using your answer in (a).

Use a Riemann sum with \(n=4\) and left endpoints to estimate the area under the graph of \(f(x)=4-x\) on the interval \(1 \leq x \leq 4\). Then repeat with \(n=4\) and midpoints. Compare the answers with the exact answer, \(4.5\), which can be computed from the formula for the area of a triangle.

Displacement versus Distance Traveled The velocity of an object moving along a line is given by \(v(t)=2 t^{2}-3 t+1\) feet per second. (a) Find the displacement of the object as \(t\) varies in the interval \(0 \leq t \leq 3\). (b) Find the total distance traveled by the object during the interval of time \(0 \leq t \leq 3\).

Displacement versus Distance Traveled The velocity of an object moving along a line is given by \(v(t)=t^{2}+t-2\) feet per second. (a) Find the displacement of the object as \(t\) varies in the interval \(0 \leq t \leq 3 .\) Interpret this displacement using area under the graph of \(v(t)\). (b) Find the total distance traveled by the object during the interval of time \(0 \leq t \leq 3\). Interpret this distance as an area.