Homework 1Solutions

Exercise 1. (a) Estimate the area under the graph of f(x) = 1 + x2 from x = −1to x = 2 using three rectangles and right endpoints. Sketch the curve and theapproximating rectangles.

(b) Repeat part (a) using left endpoints.

(c) Repeat part (a) using midpoints.

(d) From your sketches in parts (a)-(c), which appears to be the best estimate?

1

Exercise 2. Use the definition of the area under a graph as the limit of sums of approx-imating rectangles to write the area under the curve y = x3 from x = 0 to x = 1. Thenuse the following formula for the sum of the first n cubes to evaluate your limit.

13 + 23 + 33 + . . .+ n3 =

[n(n+ 1)

2

]2

Exercise 3. Use the limit definition to evaluate

∫ 5

−1(1 + 3x) dx

2

Exercise 4. Evaluate the integral by interpreting it in terms of areas.

1.∫ 0−3(1 +

√9− x2) dx

2.∫ 2−1 |x| dx

Exercise 5. If∫ 51 f(x) dx = 12 and

∫ 54 f(x) dx = 3.6, find

∫ 41 f(x) dx.

Exercise 6. Evaluate the following integrals:

1.∫ 20 (6x

2 − 4x+ 5) dx

3

2.∫ 10 x4/5 dx

3.∫ 0−1(2x− ex) dx

4.∫ 2−2(3u+ 1)2 du

5.∫ −1−2

(4y3 + 2

y3

)dy

6.∫ 10 x( 3√x+ 4√x) dx

7.∫ π/40 sec2 t dt

8.∫ 91

12x dx

4

9.∫ √3/21/2

6√1−t2 dt

10.∫ 641

1+ 3√x√x

dx

11.∫ 1−1 e

u+1 du

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