Area under a curve. Use the following steps to approximate the area under the curve f(x) = 2(x â 2) + 8 on. the interv
Math 220 Fall 2015 Worksheet October 27, 2015
Name:
Please justify all answers on a separate piece of paper.
1.
(a) Discuss in your group what it means when we say: ”f (x) is an antiderivative for g(x).”
(b) Discuss how we would find an antiderivative for a function g(x).
2.
A few interesting antiderivatives Find f for each of the following. (a) f 0 (x) = e7x
(b) f 0 (x) = 10 cos(5x)
(c) f 0 (x) =
3.
1 1−x
Playing with shapes Sketch the region whose area is given by the following integrals (a and r are positive constants). Then use geometry to evaluate the integral. Z a Z a Z r√ (a) 4 dx (b) (a − |x|) dx (c) r2 − x2 dx −a
−a
−r
4.
Area under a curve Use the following steps to approximate the area under the curve f (x) = 2(x − 2) + 8 on the interval [−1, 2]. (a) Sketch the graph y = f (x) on the interval [−1, 2]. (b) Draw 3 rectangles such that each has one side on the x-axis, the bottom sides divide the interval into equal pieces, and the height of the rectangle is equal to the function at the left endpoint. (c) Compute the total area of the rectangles from part (b). (d) Is the area of those bigger, smaller, or equal to the area under the curve?
(e) Now repeat but have the heights of the rectangles be the function at the right endpoints. Calculate the sum of the area of those rectangles. Is the area of those bigger, smaller, or equal to the area under the curve?
(f) What do you think would happen if we used 6 rectangles? 12 rectangles? 24 rectangles? How many rectangles do we want to use? (g) Can you come up with an expression to find the exact area?
5.
Consider the function f (x) = x2 + 1 on the interval [1, 3]. (a) Sketch the graph of f (x) on the interval [1, 3]. (b) Set up a Riemann sum using left endpoints which will estimate the area under the curve using four rectangles. (c) Set up a Riemann sum using left endpoints which will estimate the area under the curve using n rectangles. (d) Compute the limit of the previous sum as n goes to ∞. What should this limit tell you?
6.
Explain why the following are true. Include a picture with each explanation. Rb (a) If f (x) ≥ 0, then a f (x) dx ≥ 0. Rb Rb (b) If f (x) ≥ g(x), then a f (x) dx ≥ a g(x) dx. Rb (c) If m ≤ f (x) ≤ M, then m(b − a) ≤ a f (x) dx ≤ M (b − a).
7.
Use the inequalities in the previous problem to verify: √ √ Z π/4 2π 3π cos x dx ≤ ≤ . 24 24 π/6
8.
”Add up areas of each part to get whole area” property of integrals Let f (x) and g(x) be continuous functions on R. Suppose we know the following: Z 4 Z 4 Z 0 Z 10 f (x) dx = 12, g(z) dz = −3, f (y) dy = 7, and g(t) dt = 12. 0
0
10
Compute the following definite integrals. R4 R 10 (a) 0 (f (x) + g(x)) dx (c) 4 −3f (x) dx R 10 R 10 (b) 0 2f (x) dx (d) 4 (5 − 2g(x)) dx
