Area of a Circle (take 2)
At the beginning of the term, we saw one approach to computing the area of a circle. Here is another way to view this problem.
Consider the circle of radius 1 centered at the origin. We can represent the curve of the top half of this circle with the function .
1. Integrals in x
Let F(x) represent the shaded area under the curve between t=0 and t=x.
Use geometry to evaluate F(0) and F(1).
Is F(x) increasing or decreasing? Justify your answer.
Is F(x) concave up or down? Justify your answer.
Sketch an approximate graph of y = F(x).
Write an integral that represents the shaded area for arbitrary x. (We can’t compute this integral, yet.)
2. Geometry in
Alternatively, we can represent this region in terms of . Let represent the area shaded in the figure below.
Use geometry to evaluate and . (Hint: recall that the area of a sector of a circle with radius r
subtended by an angle a is .)
Is increasing or decreasing? Justify your answer.
Is concave up or down? Justify your answer.
Sketch an approximate graph of . What is the domain?
Use geometry to write an expression in for , the area of the shaded region for an arbitrary .
Compute the derivative .
3. Integrals in
The figure below shows the relationship between x and .
Use this figure to describe x in terms of . That is, write a formula that computes x given . (Hint: use trigonometry).Substitute this expression for x into the integral for F(x) you wrote in part 1. You now have a function described by an
integral.Use the Fundamental Theorem of Calculus to compute .
Use algebra and trigonometric identities to show that .Use this result to make a conclusion about what must be true about the function .