Let f(x) = 1 - x 2 . a. Show that the graph of f is the
Question:
Let f(x) = √1 - x2.
a. Show that the graph of f is the upper half of a circle of radius 1 centered at the origin.
b. Estimate the area between the graph of f and the x-axis on the interval [-1, 1] using a midpoint Riemann sum with n = 25.
c. Repeat part (b) using n = 75 rectangles.
d. What happens to the midpoint Riemann sums on [-1, 1] as n →∞?
Step by Step Answer:
a Note that if y 1 x 2 then y 2 1 x 2 so x 2 y 2 1 which represents a circle of ra...View the full answer
Calculus Early Transcendentals
ISBN: 978-0321947345
2nd edition
Authors: William L. Briggs, Lyle Cochran, Bernard Gillett
Related Video
The area of a circle is given by the formula: A = ?r^2 Where A is the area, ? is a mathematical constant (approximately 3.14), and r is the radius of the circle. The radius is the distance from the center of the circle to the edge. So for example, if the radius of a circle is 5 units, the area would be: A = ?(5^2) = ?(25) = 78.5 square units Here three friends went out to a restaurant and decided to have pizza for lunch. While they were waiting for their order, one of them stepped out to take a call but promised to return in two minutes. The waiter came and presented them with the menu, and they decided to order two 10-inch pepperoni pizzas. The waiter took their order and left. Just then, the third friend returned and asked what they had ordered. They explained that they had ordered two small pepperoni pizzas as they were only three of them and it would be enough for them. However, the third friend suggested that for the same price, they could get a large pizza that would be much bigger than the two small ones. He asked the waiter for a pen and paper and proceeded to calculate the area of the two small pizzas and compared it to the area of one large pizza. The friends were amazed at how the circle measurements worked and decided to change their order from two small pizzas to one large pizza.
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