INTEGRATION: AREAS AND RIEMANN SUMS MR. VELAZQUEZ AP CALCULUS

INTEGRATION: AREAS AND RIEMANN SUMS MR. VELAZQUEZ AP CALCULUS

APPROXIMATING AREA For today s lesson, we will be using different approaches to the area problem. The area problem is to definite integrals as the tangent line problem is to derivatives. To begin, let s try to determine the area between the x-axis and the function f x = x 2 + 1 from x = 0 to x = 2. This area is shown as the shaded green region below: Note: For the sake of comparison, the actual area of this region is A = 4. ത6

APPROXIMATING AREA Our strategy will be to use rectangles to approximate this area. We ll start by using a total of four rectangles, and we ll choose them so that the length of their bases are all equal. To get the length x of each base, we simply have to divide the length of the interval 0, 2 by the number of rectangles. f x = x 2 + 1 Base of each rectangle for an interval a, b divided into n rectangles: x = b a n In this case x = 2 0 = 1 or 0.5 4 2

APPROXIMATING AREA Notice that for each base, we can form a rectangle whose height is given by the value of the function for the right-hand side of the rectangle. We can now find the area of each rectangle, add them all together, and the result will be our approximate area. f x = x 2 + 1 A = 1 2 f 1 2 + 1 2 f 1 + 1 2 f 3 2 + 1 2 f 2 A = 1 2 5 4 + 1 2 2 + 1 2 13 4 + 1 2 5 A = 5. 75

APPROXIMATING AREA Of course we don t necessarily have to choose the value of the function at the right-endpoint side of each rectangle. Below are two alternatives. Left-endpoint Midpoint Note that we end up with different results with each method. Sometimes we will clearly have an overestimate, sometimes the result will be an underestimate.

APPROXIMATING AREA The easiest way to get a better approximation is to simply increase the number of rectangles. Or in other words, increase the value of n. Here are the results if we double the number of rectangles from 4 to 8. A = 5.1875 A = 4.1875 A = 4.65625

EXAMPLE: Estimate the area between f x = x 3 5x 2 + 6x + 5 and the x-axis using n = 5 rectangles, assuming right-endpoint function height.

EXAMPLE: Estimate the area between f x = x 3 5x 2 + 6x + 5 and the x-axis using n = 5 rectangles, assuming left-endpoint function height.

EXAMPLE: Estimate the area between f x = x 3 5x 2 + 6x + 5 and the x-axis using n = 5 rectangles, assuming midpoint function height.

RIEMANN SUMS We now use sigma notation to simplify our notation a little. n A f(x i ) x i=1 Summation symbol. We are adding terms where the first term uses i = 1 and the last term will use i = n. A value of x which lies in each interval. Can be the rightendpoint, leftendpoint, midpoint, or none of these. Base of each rectangle. Easier if they are all equal, but they don t have to be. The summation above is known as a Riemann Sum.

RIEMANN SUMS As previously seen, we can increase our accuracy by increasing n, the number of rectangles. Therefore, if we allow n to go to infinity we will get the exact area. Not an estimate. n A = lim n i=1 f(x i ) x Remember this, as it is very closely tied to the definition of a definite integral.

EXAMPLE: Estimate the area under the graph of g x = 2x + 12, x [0, 6] using n = 6 rectangles and letting x i equal the midpoint of each subinterval. Write the Riemann sum used in this approximation.

EXAMPLE: Estimate the area under the graph of h x = cos x, x π, π using n = 4 2 2 rectangles and letting x i equal the left endpoint of each subinterval. Write the Riemann sum used in this approximation.

EXAMPLE: PHYSICS Suppose a car travels along a straight line with a velocity function in meters/sec given by: v t = t 2, t 0, 10 with t measured in seconds. Use a Riemann sum to estimate the distance traveled by the car on this interval. Approximate this using n = 5 and letting x i equal the right endpoint of each subinterval.

HOMEWORK & CLASSWORK MATH JOURNAL: Don t forget to write down what you learned! CLASSWORK: RIEMANN SUMS: Estimate the following areas using a Riemann sum, using the given number of rectangles. Use your preferred method for the function values (leftendpoint, right-endpoint or midpoint). f x = x 3 2x 2 + 4 on [1, 4] with n = 6 g x = 4 x 2 + 2 on 1, 3 with n = 8 h x = x cos x 3 on 0, 3 with n = 6 HOMEWORK: Calculus of a Single Variable (old book), Pg. 268, #27-32 (all)