Calc II Midterm 1

Integration

100

∫2xcos(x2)dx

sin(x²)+C

200

∫1/(xlnx)dx

ln∣lnx∣+C

300

∫xe^{(x^2)}dx

1/2e^x^2+C

400

∫x/(sqrt(1+x²))dx

sqrt(1+x^2)+C 

500

∫sin³xcosxdx

(sin⁴x)/4 +C

Disks & Washers

100

the volume when y=x², from 0 to 1, rotated about the x-axis.

π/5

200

the volume when y=x², from 0 to 1, rotated about the y-axis

π/2

300

Region between y=x and y=x², rotated about x-axis

2π /15

400

the washer setup

& the washer setup key

V=π∫(R²−r²)dx

outer radius − inner radius

500

Switch to x=sqrt(y)

V=π∫(R²−r²)dy

Shells & Slicing

100

When should you use shells instead of washers?

when slicing parallel to axis of rotation or when solving for the other variable is messy.(draw picture)

200

Rotate y=x², 0→1, about y-axis (shells)

π/2  

300

shell setup

V=2π∫(radius)(height)dx

400

slices impact

• Perpendicular slices → disks/washers
• Parallel slices → shells 

500

Compare slicing vs shells; which is easier and why?

• function is already in terms of x
• avoids solving for inverse 
• when slicing parallel to axis of rotation 
• when solving for the other variable is messy.

Work (Springs & Ropes)

100

Hooke's Law (spring) F=

F=kx

200

Spring stretched 2m with force 10N → find k

5

300

Work to stretch a spring (k=5) from 1m to 3m

20

400

A rope of length 10 m hangs over the side of a building. The rope has a mass density of 2 kg/m. How much work is required to pull the entire rope up to the top?

980 J

500

A bucket containing 20 kg of water is lifted 10 m to the top of a building. The bucket itself weighs 5 kg. As it is lifted, water leaks out at a constant rate, so that by the time it reaches the top, the bucket is empty. How much work is done?

1470 J

Inverse + Log/Exp

100

Derivative of ln⁡x

 ∫e^xdx

1/x

e^x+c

200

Derivative of e^3x

∫1/xdx 

3e ^3x

ln∣x∣+C

300

(f−1)′(a)

∫e^2xdx

1/(f′(f−1(a)))

1/2e^2x+C 

400

Differentiate: y=ln⁡(x²+1)

2x/(x²+1)

500

Differentiate: y=x^x 

x^x(lnx+1)