Category 1: Antiderivatives

Category 1: Antiderivatives

Week 6 Lab

Name______

Part 1 Directions: For each category see if you can devise the rule(s) that was used to find the antiderivative/indefinite integral and then use it to find the antiderivative/indefinite integral of te functions that follow.

Category 1: Antiderivatives

  1. Function:

Antiderivative:

  1. Function:

Antiderivative:

  1. Function:

Antiderivative:

  1. Function:

Antiderivative:

What rule or process is used to find an antiderivative? How is an antiderivative related to a derivative?

The antiderivative uses the reverse process that was used to determine the derivative. For a power the one is added to the exponent and the entire term is divided by the new exponent value.

Find the antiderivatives of the following functions.

  1. 3.
  1. 4.

Is the only possible antiderivative of? If not, why not?

No, a constant could have been added to the cube of x, as the derivative of a constant is 0.

How can I write a function that represents all possible antiderivatives of a given function?

F(x) + c

Category 2: Indefinite Integral ()

  1. Integral:

Solution:

  1. Integral:

Solution:

  1. Integral:

Solution:

  1. Integral:

Solution:

How is the process of integration related to finding an antiderivative? How is it different? How are the solutions arrived at?

The process is similar, with the exception of adding a constant to the solution to account for all possible solutions.

Solve the following integration problems

  1. 3.
  1. 4.

If a(t) = v’(t)=s”(t) how can I use integration to solve the following problem? What is the solution?

If a space shuttle’s downward acceleration is given by a(t)= -32 ft/s2 and its initial velocity is v(0) = -100 ft/s and its initial position is s(0) = 100,000 ft, then after how many seconds will it hit the ground?

Part II - Directions: For the function we are going to estimate the area under the curve in the interval [0, 2] by do the following:

  1. Divide the interval into 5 equal pieces. How long is each piece? What will the x-values be for the endpoint of each piece? This will represent the width of the rectangles we will use to estimate the area in gray above.

Width of rectangle =

0

0.4

0.8

1.2

1.6

2.0

  1. Evaluate f(x) for x1 to x5 the left-endpoints. This will represent the height of the rectangles we will use to estimate the area under the curve.
  1. Using the heights from part 2 and the width from part 1 draw the rectangles onto the graph. Be sure that you measure the height from each left endpoint.

Based on the sketch will the estimate you get from the left endpoints be an over or underestimate? Why?

Based on the sketch above this method will overestimate the area under the curve, because the boxes contain more area than just the area under the curve.

  1. Find the area of each of the five rectangles and add them together to get your estimate? What is it?
  1. Repeat steps 2 through 4 again for the right endpoints x2 to x5

Based on your sketch will using the right endpoints provide an over or underestimate? What is the estimate?

Based on the sketch this method will underestimate the area under the curve, because the rectangles do not cover all of the area under the curve.

We have used a method for estimating area under the curve that sums the areas of a series of rectangles we can symbolically write that as

If you wanted to improve the estimates you get from summing the areas of rectangles what could you do?

Add many more rectangles, the more rectangle I add the better the estimate would be.

Symbolically we can write this as = = The Definite Integral = the exact area under the curve.

  1. Estimate the area under the curve in the interval [-3, 3] using n = 6 rectangles with left endpoints.
  1. Estimate the area under the curve in the interval [0, 4] using n = 4 rectangles with right endpoints.

Part 3

Category 1: The Definite Integral and the Fundamental Theorem of Calculus

1. Integral:

Antiderivative:

Rule:

2. Integral:

Antiderivative:

Rule:

3. Integral:

Antiderivative:

Rule:

4. Integral:

Antiderivative:

Rule:

How is the Antiderivative of the given integral being used to evaluate the given definite integrals? What is the rule being used to evaluate the given definite integrals? (Hint: This rule is the single most important rule in all of Calculus.)

The Fundamental Theorem of Calculus is used to evaluate the integral after the antiderivative is calculated.

Why can’t you find the value of this definite integral? .

The power of a positive number is equal to the power of a negative number, in this case, and therefore the answer, when the definite integral is centered on zero, like this one, is always zero.

Evaluate these definite integrals

1. 3.

2. 4.

Top View