15.2 Integration over General Regions

Integration

100

The integral of -sin(x)

What is cos(x)?

200

The integral of sec²(y)?

What is tan(y)?

300

The integral of (1-(1/w)) * cos(w - ln(w)) dw

What is sin(w - ln w) + c?

400

The integral of 1/(1+x²)

What is tan^-1(x)?

500

The integral of xe^6x

What is (x/6)e^6x - (1/36)e^6x + c?

(integration by parts)

General Bounds

100

The iterated integral for the double integral over the described region R, in either order (GB1)

Either:

What is 0</=x</=2 and x³</=y</=8

What is 0</=y</=8 and 0</=x</=y^1/3

200

The iterated integral for the double integral over the described region R, in either order (GB2)

What is either

0<x<3, x²<y<3x or

0<y<3, y/3<x<root(y)

300

The iterated integral for the double integral over the described region R, in BOTH orders (GB3)

What are

0<x<2, 1<y<e^x

1<y<e², ln(y)<x<2

400

The setup for the iterated integral for the double integral of f(x,y)=x²  over the described region R, in BOTH orders. R is the area bounded by y=3-2x, y=x, and x=0.

What are 

0<x<1, x<y<3-2x

and the split integral

[0<y<1, 0<x<y] +[1<y<3, 0<x<(1/2)(3-y)]

500

The setup for the double integral of z=16-x²-y² over the described region R, in BOTH orders. R is the area bounded by y=2x^1/2, y=4x-2, and y=0.

What are

0<y<2, y²/4<x<(y+2)/4

and the split integral

[0<x<0.5, 0<y<2x^1/2] + [0.5<x<1, 4x-2<y<2x^1/2]

Order of Integration

100

The equivalent integral with a reversed order of integration for OI1.

What is 0<y<4, 1<x<2?

200

The equivalent integral with a reversed order of integration for OI2.

What is 2<y<4, 0<x<(4-y)/2?

300

The equivalent integral with a reversed order of integration for OI3.

What is 0<x<1, x²<y<x?

400

The equivalent integral with a reversed order of integration for OI4.

What is -1<x<1, 0<y<(1-x²)^1/2?

500

Write 6 different iterated triple integrals for the volume of the rectangular solid in the first octant bounded by the coordinate planes and the planes x=1, y=2, and z=3.

What is OI5?

Region of Integration

100

The sketch of the region of integration for RI1.

What is the sketch of RI1?

200

The sketch of the region of integration for RI2.

What is the sketch of RI2?

300

The sketch of the region of integration for RI3.

What is the sketch of the region of RI3?

400

Sketch of the region of integration for the integral of f(x,y)=x/y over the region in the first quadrant bounded by the lines y=x, y=2x, x=1, and x=2.

What is RI4?

500

Sketch and evaluation of the integral for RI5

What is the sketch and evaluation of RI5?

Misc

100

The room we are in 

What is Cotchett Edu 203?

200

The date of Exam 2.

What is November 7th?

300

x and y in polar coordinates

What are 

x = r cos(theta)

y = r sin(theta)

400

Find the volume of the solid cut from the first octant by the surface z=4-x²-y

What is 128/15?

500

Molly's favorite color.

What is blue?