Anti-derivatives and Areas Under Curves Review

U-Substitution to Integrate Functions

In order to integrate using U-substitution, there are several steps that must be followed.

1) Set u equal to a part of the problem.

2) Find the derivative of . This is du.

3) Use substitution to remove all variables besides u from the problem. If there are still other variables, solve for the variable using the definition of u and plug that in.

4) If there are limits, use the definition of u to find the limits with respect to u.

5) Find the anti-derivative.

6) (NO LIMITS) Substitute for u so that the answer is in terms of the original variable.

6) (LIMITS) Evaluate u at the upper and lower limits. Then subtract them.

For me, the hardest part of U-substitution is remembering to change the limits and finding the anti-derivative.

Example:  Integrate the function f(x) = x / √(2x-5) dx on [3, 7]

u = 2x - 5           Find u

du = 2dx            Find du

x dx / √u            Substitute u

1/2 | 2dx x / √u  Plug in a two and pull out a 1/2

1/2 | du x / √u    Substitute du

u = 2x - 5
2x = 5 + u
x = (5 + u) / 2    Solve for x and substitute it in

1/2 | du ((5 + u) / 2) / √u

u = 2(3) - 5

u = 1                  Find the new lower limit

u = 2(7) -5

u= 9                   Find the new upper limit
[1, 9]

1/4 | (5 + u)(u^-1/2)    Rewrite the equation so it is easier to find the antiderivative
1/4 | 5u^-1/2 + u^1/2

1/4 | 10u^1/2 +(2/3)u^3/2          Find the anti-derivative

1/4 (30 + 18) - 1/4 (10 + 2/3) Plug in the limits and solve for the answer.

1/4(48) - 1/4 (32/3)

12 - (8/3)

28 / 3      

Fundamental Theorem of Calculus Part II to Find the Area Under a Curve
The FTC part II says that if f(x) on [a, b] then f(x)dx on [a, b] = F(b) - F(a)
In order to do this, there are some steps that must be followed.
1) Find the anti-derivative of the function.
2) Plug the upper limit into the anti-derivative.
3) Plug the lower limit into the anti-derivative.
4) Subtract the two for the answer.

For me, the hardest thing to remember for this learning target is to always take the anti-derivative and not the derivative by mistake.

Example: Find the area under the curve of f(x) = 4 - 2x² on [0, 12]

4x - (2x / 3)³ + C                                Find the anti-derivative

(4(12) - (2/3)12³) - (4(0) - (2/3)(0)³) Plug in the upper and lower limits

(48 - 1152) - 0                                    Simplify

-1104                                                  This is your answer

Area Under a Curve Using Left and Right Endpoints
In order to do this, there are several steps that must be followed.
1) Determine the width of your rectangles. The more rectangles you use, the more accurate the area will be.
2) Set up a summation the properly shows how you are solving the problem. When you write the summation, the n is the number of rectangles you are using and it goes at the top. The i shows what point you are starting with and it goes at the bottom. To the right of the summation is Δx f(lower limit + iΔx). Δx = interval / number of rectangles
3) Solve the summation using either the left or right endpoints.

For me, the hardest part of these problems is to mess up which points to use for left or right.

Example: Approximate the area under the curve using 6 rectangles for the function
f(x) = 4 - 2x² on [0, 12]

Using Right Endpoints

n = 6 i = 1 Δx = 12 / 6 = 2

  6
  Σ 2(0 + 4 - 2(2i)²)
i = 1

  6
  Σ 2(4 - 2(2i)²)
i = 1

2(-4 - 28 - 68 - 124 - 196 - 284)

2(-704)

-1408 is the answer using right endpoints

Using Left Endpoints

  5
  Σ 2(0 + 4 - 2(2i)²)
i = 0

  5
  Σ 2(4 - 2(2i)²)
i = 0

2(4 - 4 - 28 - 68 - 124 - 196)

2(-416)

-832 is the answer using left endpoints