AP Calc Unit 8: Applications of Integrals

Average Value Theorem

Integral of Acceleration

Velocity

Integral of Velocity

Position

Total Distance

∫|v(t)| dt

• w/o absolute value: total displacement

How do you find final value?

Final Value = Initial + Change

s(b) = s(a) + ∫ v(t) dt

Area Between Curves: In terms of x

Area Between Curves: In terms of y

∫ (f(y) - g(y)) dt

• right-left

How do you find cross sections?

• Use the area formula based on the shape of the cross section

• Use f(x)-g(x) as your s/r

Cross Sections: Area of Squares

s²

Cross Sections: Area of Triangles

½ bh

Cross Sections: Area of Semicircles

½πr²

• r= f(x)-g(x)/2

Cross Sections: Area of Equilateral Triangles

√3/4s²

Cross Sections: Area of Rectangles

bh

Disk Method

• Used with only ONE function: when it touches the x-axis

Washer Method

• Used with TWO functions: when space b/t x-axis

Cross Sections: Isosceles Right Triangle

½ L²