Given a circular sector with radius L and central angle $\theta$, show that the area of the sector is given by

$$S=\frac{1}{2} L^2 \theta .$$

Let R be the region bounded by y=1 / x, the x-axis, x=1, and x=b, where b>1. Let D be the solid formed when R is revolved about the x-axis.
Show that $S \rightarrow \infty$ as $b \rightarrow \infty$.

Let R be the region bounded by y=1 / x, the x-axis, x=1, and x=b, where b>1. Let D be the solid formed when R is revolved about the x-axis.
Show that $V$ approaches a finite limit as $b \rightarrow \infty$.

Let R be the region bounded by y=1 / x, the x-axis, x=1, and x=b, where b>1. Let D be the solid formed when R is revolved about the x-axis.
Write the surface area $S$ as an integral.

Let R be the region bounded by y=1 / x, the x-axis, x=1, and x=b, where b>1. Let D be the solid formed when R is revolved about the x-axis.
Find the volume $V$ of $D$.

Use integration to find the area of the figure having the given vertices. \

$$(2,-3),(4,6),(6,1)$$

For the metal sphere in the previous Exercise , let R=5. What value of r will produce a ring whose volume is exactly half the volume of the sphere?

Solve the differential equation. \

$$\dfrac{d y}{d x}=6-x$$

Find the accumulation function F. Then evaluate F at each value of the independent variable and graphically show the area given by each value of F. \

$$F(y)=\int_{-1}^y 4 e^{x / 2} d x$$

\

$$F(4)$$

Think About It Consider the equation $\frac{x^2}{9}+\frac{y^2}{4}=1$.
Compare the interval of integration in the previous part and the domain of the integrand. Is it possible to evaluate the definite integral? Is it possible to use Simpson's Rule to evaluate the definite integral? Explain.

Find the accumulation function F. Then evaluate F at each value of the independent variable and graphically show the area given by each value of F. \

$$F(y)=\int_{-1}^y 4 e^{x / 2} d x$$

\

$$F(0)$$

Think About It Consider the equation $\frac{x^2}{9}+\frac{y^2}{4}=1$.
Set up the definite integral for finding the first quadrant arc length of the graph in the previous part.

Find the accumulation function F. Then evaluate F at each value of the independent variable and graphically show the area given by each value of F. \

$$F(y)=\int_{-1}^y 4 e^{x / 2} d x$$

\

$$F(-1)$$

Think About It Consider the equation $\frac{x^2}{9}+\frac{y^2}{4}=1$.
Use a graphing utility to graph the equation.