Math 205 Midterm

Math 205 Midterm

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1. (5 marks): (a)	Approximate the de nite integral A =					3
		R			approximating rectangles of equal widths.
	Riemann sum		3	using 3		R

(b)Now nd the approximation by a left Riemann sum L3, again using 3 approximating rectangles of equal widths.

(c)Calculate the exact value of A by integrating. Comparing this value with

12 (R3 + L3) : Do your results found in (a) and (b) appear reasonably

accurate?

2.

(3 marks): (a) Use Part 1 of the Fundamental Theorem of Calculus to nd F0(x)

3

p

dt.

for F(x) =

cos x

1 + t3

R

10t

3.

(5 marks): Find f(t); given f0(t) =

p3

and f(8) = 20.

t 2

4.

(10 marks): Calculate the following inde nite integrals

Z

p

x3

Z

3

x2 + 4x + 4

(a)

dx

(b)

dx

x3 + x

16 x2

5.

(12 marks): Evaluate the following de nite integrals (do not approximate):

e

(a)

Z0

cos4 x tan2 x dx

(b)

Z1

x2 ln xdx

6. (5 marks): Find the area of the region enclosed by the curves x = jyj and x = y2 2.

Bonus. (2 marks): Given that

Z

[f (x) + f00 (x)] sin xdx = 2

0

and f ( ) = 1; nd f (0).