Midterm 1 Study Guide - Stewart 7th Ed PDF

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Math 115 – Midterm I Study Guide Sections 4.9-6.3

1. Evaluate the integral, if it exists. 2

∫ ( 8 x 3 +3 x 2 ) dx a.

1

1

∫ (1−x 9 )dx b.

0

9

∫√ c.

1

u−2 u2 du u

1

d.

∫ y ( y 2+1 )5 dy 0

5

e.

dt

∫ (t−4 )2 1

1

f.

∫ v 2 cos( v 3 )dv 0

π 4

4

t tan t dt ∫ 2+cos t

g.

−

π 4

3

h.

∫|x 2−4|dx 0

2. Evaluate the integral, if it exists.

a.

∫(

)

2

1−x dx x

b.

x+2 dx 2 x +4 x √

c.

∫ sin πt⋅cosπt⋅dt

d.

∫ √ x dx

∫

e√ x

e.

∫ tan x⋅ln(cos x)dx

f.

∫ 1+x x4 dx

3

g.

∫

sec θ⋅tanθ dθ 1+secθ

cos x

h.

∫ √ 1+sin x dx

3

3. Use the Midpoint Rule with n = 6 to approximate

∫ sin( x 3 )dx 0

4. Find the area of the region bounded by the given curves. a.

y = x 2 , y=4 x −x 2

b.

y =1 −2 x 2 , y=|x|

c.

y=√ x , y=x 2 , x =2

5. Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. a.

y =2 x , y= x 2 ; about the x-axis

b.

x =1+ y 2 , y= x −3 ; about the y-axis

c.

x =0, x =9− y 2 ; about x = -1

d.

y =1+ x 2 , y=9−x2 ; about y = -1

6. Review from the worksheets for Riemann sum

Answer Key: 1. a) 37

9 10

b) c) -76

21 4

d) e) Does not exist

1 sin 1 3

f) g) 0

23 h) 3

2. a)

1 − −2 ln|x|+x +C x

b)

√ x2 +4 x + C

c)

1 ( sin πt )2 +C 2π

d)

2 e√ x +C

1 − (ln(cos x ) ) 2 +C 2 e) 1 ln|1+x 4 |+C f) 4 g)

ln|1+secθ|+C

h)

2 √ 1+sin x +C

¿ 0 .28098 8 4. a) 3 7 b) 12 10 4 − √2 c) 3 3 3.

64 π 5. a) 15 117 π 5 b) 1656 π 5 c) d)

256 π...