Warm up Problems

1.

2.

3.

35 4sinx x dx

2

1 1 dxx x

7 xe dx

More With IntegralsIt can be helpful to guess and adjustEx.

Ex.

3xe dx

102 1x dx

Ex.2 1t dtt

Thm. Fundamental Theorem of CalculusIf f (x) is a continuous function on [a,b], and if F(x) = f (x), then

- F(x) is an antiderivative of f (x).

- Find an antiderivative, then plug in the endpoints

b

a

f x dx F b F a

Ex.

Ex.

5

3

2x dx

2

0

sin x dx

Ex.

Pract.

2

0

xe dx

24

1

x dx

- Don’t write “+ c” on definite integrals- We could use a calculator to get the

answer, but this way we get the exact answer, not just a decimal approximation

Ex. Let R be the region bounded by ,

the x-axis, x = 1, and x = 4. Find a value for h so that the line x = h splits R into two regions of equal area.

1yx

Def. A differential equation is an equation that involves a function and some of its derivatives.

Ex. y = 3y - 5y2

Ex. Let , find the general solution.5dy x xdx

Ex. Solve the IVP if y(1) = 3.1dy

dx x

Def. An initial value problem (IVP) is a differential equation and a value of the solution.