SECTION 5.4The Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus

• Basically, (definite) integration and differentiation are inverse operations.

Example 1Evaluate the definite integral. Verify your result with a graphing calculator.

Example 2Evaluate the definite integral. Verify your result with a graphing calculator.

Example 3Evaluate the definite integral. Verify your result with a graphing calculator.

Example 4Evaluate the definite integral. Verify your result with a graphing calculator.

Example 5Evaluate the definite integral. Verify your result with a graphing calculator.

Example 6Evaluate the definite integral. Verify your result with a graphing calculator.

Example 7Determine the area of the given region.

Example 8 Find the area of the region bounded by the graphs of the equations.

The Mean Value Theorem for Integrals

• We know the area of a region under a curve is . . . 1. greater than the area of an inscribed rectangle, &

2. less than the area of a circumscribed rectangle.

• The MVT states that there exists some rectangle “between” those with area equal to the area of the region under the curve.

The Mean Value Theorem for Integrals (cont.)

Average Value of a Function• The value given in the MVT is called the average value of on the interval

Example 9 Find the average value of on the interval and find the values of where the function equals its average value.

Second Fundamental Theorem

Example 10 Use the Second Fundamental Theorem of Calculus to find .