Exam 2

Question 1

How to find area between two curves

Answer 1

1) Graph the functions
2) Find the intersection point by setting both functions equal to 0
3) Set up the integral from a to b (a being the left bound placed on bottom of integral and b being the right bound placed on top of the integral)
4) The integral inside is the top function minus the bottom function dx
5) Integrate and then plug in b and subtract plugged in a

Question 2

How to find area between two curves in terms of y

Answer 2

1) Rewrite the function from y= to x=
2) Find the intersection point by setting both functions equal to 0
3) Set up the integram from a to b with a being the lowest bound on the bottom of integral and b being the highest bound on the top of the integral
4) The integral inside is the right function minus the left function dy
5) integrate and then plug in b and subtract plugged in a

Question 3

How to find volume w/ cross sections (semicircle)

Answer 3

integral from a to b 1/2*pi((f(x)-g(x))/2)^2 dx
= 1/2pi int from a to b ((f(x)-g(x))/2)^2 dx

Question 4

How to find volume w/ cross sections (squares)

Answer 4

int from a to b (f(x)-g(x))^2 dx

Question 5

How to find volume w/ cross sections (rectangle)

Answer 5

int from a to b k*(f(x)-g(x))^2 dx

(height is k of length)

Question 6

How to find volume w/ cross sections (circle)

Answer 6

int from a to b pi(1/2(f(x)-g(x))^2 dx

Question 7

How to find volume w/ cross sections (isosceles right triangle)

Answer 7

int from a to b 1/2(f(x)-g(x))^2 dx

Question 8

How to find volume w/ cross sections (equilateral triangle)

Answer 8

int from a to b (sqrt of 3)/2*(f(x)-g(x))^2 dx

Question 9

Disk method

Answer 9

int from a to b pi(f(x))^2 dx

Question 10

Washer method

Answer 10

int from a to b pi[(R(x))^2-(r(x))^2] dx

big R is the outer radius (or top function)

Question 11

Shell method

Answer 11

int from a to b 2pixf(x) dx

Question 12

Arc length

Answer 12

int from a to b sqrt: 1+(f’)^2 dx

Question 13

Surface area

Answer 13

int from a to b 2pif(x)sqrt: 1+(f’)^2 dx

Question 14

Find work of movement

Answer 14

int from 0 to h (weight) dx

Question 15

Find work of spring

Answer 15

int from a to b k*x dx

F=kx
F= force, x=distance adjusted