Calculus II (MAT 146)Dr. Day Wednesday January 21,

2015

Quiz #3

6.1 Area Between Curves: Examples, Extensions, Questions

6.5 Average Value of a Function

Assignments

Wednesday, January 21, 2015

MAT 146

MAT 146

Area Under A Curve

(i) Calculate the area under the graph of y = x2 + 2 for 0 ≤ x ≤ 1. Include a sketch.

(ii) Calculate the area under the graph of y = 1 – x for 0 ≤ x ≤ 1. Include a sketch.

Wednesday, January 21, 2015

MAT 146Wednesday, January 21, 2015

MAT 146Wednesday, January 21, 2015

MAT 146Wednesday, January 21, 2015

1. Calculate the area between the graphs of y = 2x3 – 1 and y = x – 1 for 1 ≤ x ≤ 2.

2. Calculate the area between the graphs of y = (x–1)2 and y = 3 – x.

3. Calculate the area between the graphs of y = x3 –3x + 2 and y = x + 2.

4. Calculate the area between the graphs of x = y2 –1 and x = 3.

Area Between Curves

MAT 146

Area Between Curves

Calculate the area between the graphs of y = 2x3 – 1 and y = x – 1 for 1 ≤ x ≤ 2.

Wednesday, January 21, 2015

MAT 146Wednesday, January 21, 2015

MAT 146

Area Between Curves

Calculate the area between the graphs of y = (x–1)2 and y = 3 – x.

Wednesday, January 21, 2015

MAT 146Wednesday, January 21, 2015

MAT 146

Area Between Curves

Calculate the area between the graphs of y = x3 –3x + 2 and y = x + 2.

Wednesday, January 21, 2015

MAT 146Wednesday, January 21, 2015

MAT 146

Area Between Curves

Calculate the area between the graphs of x = y2 –1 and x = 3.

Wednesday, January 21, 2015

MAT 146Wednesday, January 21, 2015

MAT 146Wednesday, January 21, 2015

MAT 146

Strategies: Area Between Two Curves

Graph the functions in question and identify the number of bounded regions as well as which function is greater than the other for each region.

Determine the x-axis intervals (or y-axis intervals) for the bounded regions. The interval endpoints may be explicitly stated or can be determined using algebraic techniques, most typically by setting the two functions equal to each other.

Draw in a typical rectangle and determine its area. This provides essential information for the area integral you need to create.

For each bounded region, create a definite integral to represent the sum of the areas of an infinite number of typical rectangles. Evaluate this integral to determine the area of each bounded region.

Note that your TI-89 or other CAS can be a useful tool for several components of your solution process.

Wednesday, January 21, 2015

MAT 146

Area Between Curves

(A) Calculate the first-quadrant area between the graphs of y = √x and y = x2. Show a picture of the enclosed region.

(B) Set up one or more definite integrals to represent the finite area of the region enclosed by the graphs of y = 4x + 16 and y = 2x2 + 10 for−2 ≤ x ≤ 5. Do not calculate!

(C) Determine the exact area of the region enclosed by the graphs of x = −y2 + 10 and x = (y – 2)2. Sketch a graph of the region.Wednesday, January 21, 2015

MAT 146

Average Values

Question: What was the average temperature between midnight and noon yesterday?

Wednesday, January 21, 2015

MAT 146

If all we know is that it was 5º F. at midnight and that it was 21º F. at noon, the average temperature is:

Wednesday, January 21, 2015

MAT 146

If we also know that it was 16º F. at 6 am, we can re-compute an average temperature:

Wednesday, January 21, 2015

MAT 146

What if we know hourly readings?

Wednesday, January 21, 2015

12 midnight 5° F 4 am 13° F 9 am 18° F

1 am 7° F 5 am 14° F 10 am 19° F

2 am 9° F 6 am 16° F 11 am 20° F

3am11° F 7 am 17° F 12

noon 21° F

8 am 17° F

MAT 146Wednesday, January 21, 2015

MAT 146

What if we know temperatures

every minute? Every second?

Wednesday, January 21, 2015

MAT 146

The average value of a function f, on a ≤ x ≤ b, with f continuous on that interval, is:

Wednesday, January 21, 2015

MAT 146

Determine the average value of y = x2 on [0,3].

Wednesday, January 21, 2015

MAT 146Wednesday, January 21, 2015

MAT 146

Determine the average value of the function here, for the specified interval. Determine a value c such that f(c) generates that average value.

Wednesday, January 21, 2015

MAT 146Wednesday, January 21, 2015

MAT 146

WebAssign 6.1 due TONIGHT (1/21) WebAssign 6.5 due Thurs (1/22) WebAssign : Calc I Reviews due later (9/1 and 9/9)

First WA Quiz!Quiz #4 this weekend:

(5.5, 6.1, 6.5)

Assignments

Wednesday, January 21, 2015