4.1 Areas and Distances. The Area Problem: GOAL: Find the area of the region S that lies under the curve y = f(x) from a to b.

Transcription

1 4.1 Areas and Distances The Area Problem: GOAL: Find the area of the region S that lies under the curve y = f(x) from a to b. Easier Problems: Find the area of a rectangle with length l and width w. Find the area of a triangle with base b and height h. Find the area of the n-sided polygon. 1

2 Recall: When we defined a tangent, we first approximated the slope of the tangent line by slopes of secant lines and then we took the limit of these approximations. We will do something similar to find the area of a region S. Finding the Area of a Region S: 1. Approximate the region S by rectangles. 2. Take the limit of the areas of these rectangles as the number of rectangles increases. Example 1. Use rectangles to estimate the area under y = x 2 from 0 to 1. Approximate the area with four rectangles: 2

3 So the area A of S is bounded by Approximate the area with eight rectangles: So the area A of S is bounded by We can obtain better estimates by increasing the number of strips. (At this point, though, using a calculator or computer becomes a welcome tool.) The table below shows the approximations using n rectangles: n L n R n By estimating the limit of L n as n and R n as n, we can estimate the area A of the region S to be. 3

4 Example 2. For the region S bounded by y = x 2 from 0 to 1, show that the sum of the areas of the upper approximating rectangles approaches 1, that is, 3 lim n R n = 1 3 We could also show that lim n L n = 1 3. It appears that as n increases, both L n and R n become better and better approximations to the area S. Therefore we define the area A to be the limit of the sums of the areas of the approximating rectangles: Note: It is not always the case that L n < R n. 4

5 Applying the ideas of the previous examples, we can find the area of a general region S as follows: Finding the Area of a Region S: 1. Subdivide S into n rectangles of equal width. The width of the interval [a, b] is b a, so the width of each of the n rectangles is These rectangles divide the interval [a, b] into n subintervals: and 2. Find the area of the ith rectangle: A i = f(x i ) x. (f(x i ) is the value of f at the right endpoint. To find the area using the left endpoint, use f(x i 1 ) instead.) 3. Sum the areas A i from i = 1 to i = n. (Because we used the right endpoint of each subinterval, this is the right sum.) 4. Repeat the process letting n become larger and larger. Find the limit of R n as n. The A of the region S that lies under the graph of the continuous function f is the limit of the sum of the areas of approximating rectangles: If f is continuous, the limit defined above always exists. We also get the same value if we use left endpoints instead of right endpoints. In fact, we can use any value x i we want in each subinterval [x i 1, x i ]. We call the numbers x i the. So a more general expression for the area of S is Note: A is the unique number that is smaller than all the upper sums and bigger than all the lower sums. The is formed by choosing the sample points x i so that f(x i ) is the minimum value of f on the ith subinterval. The is formed by choosing the sample points x i so that f(x i ) is the maximum value of f on the ith subinterval. 5

6 We often use notation, the area of S can be written as: to write sums with many terms more compactly. Using this (Left sums and general sums can be written similarly.) In addition, the formula we used in Example 2 can be written as follows: Example 3. Let A be the area of the region that lies under the graph of f(x) = x x between x = 4 and x = 7. a) Using right endpoints, find an expression for A as a limit. Do not evaluate the limit. b) Estimate the area by taking the sample points to be midpoints and using four subintervals. 6

7 The Distance Problem: GOAL: Find the distance traveled by an object during a certain time period if the velocity of the object is known at all times. An Easy Problem: You are driving a car at a constant speed of 60 mph for 3 hours. How far did you travel? Note: We could consider this as an area: Realistically, a vehicle does not travel at a constant speed (even when on cruise control). What can we do then? Example 4. Speedometer readings for a motorcycle at 12-second intervals are given in the table. t (s) v (ft/s) a) Estimate the distance traveled by the motorcycle during this time period using the velocities at the beginning of the time intervals. b) Estimate the distance traveled by the motorcycle during this time period using the velocities at the end of the time intervals. 7

8 Notice that the calculations done in the previous example are very similar to the sums we used earlier to estimate area. In general, suppose an object moves with velocity v = v(t)., where a t b and v(t) 0 (so the object is always moving in the positive direction). We take n velocity readings at equally spaced times. Then t = b a n. During each time interval [t i 1, t i ], the velocity can be approximated by v(t i 1 ). Thus the total distance traveled during the time interval [t i 1, t i ] is approximated by v(t i 1 ) t. Thus, the distance traveled during the time interval [a, b] is estimated to be [Note: This will give a left-hand estimate of the distance traveled. To find a right-hand estimate, use v(t i ).] The more frequently we measure the velocity, the more accurate our estimates become. Thus, it seems plausible that the exact distance d traveled is Notice this is the same form as our expressions for area. Thus There are other practical ways that areas can be interpreted, as we will see throughout the following sections. Example 5. The velocity graph of a car accelerating from rest to a speed of 120 km/h over a period of 30 seconds is shown. Estimate the distance traveled during this period. 8

9 Example 6. Determine a region whose area is equal to lim n limit. n i=1 ( i ) 10. Do not evaluate the n n 9