5.1 – Estimating with Finite Sums

The Area Problem

Area Problem Our idea, as with derivatives earlier, is to use a limit of approximations to arrive at the true area. We… First approximate the region S by rectangles and then Take the limit of the areas of these rectangles as we increase the number of rectangles

Example 1 Use rectangles to estimate the area under the parabola y = x2 from 0 to 1.

Example 1 (solution)

Definition We define the area A to be the limit of the sums of the areas of the approximating rectangles: The width of each of the n strips is These strips divide the interval [a,b] into n subintervals [x0,x1], [x1,x2], [x2,x3], …, [xn-1,xn] where x0 = a and xn = b.

Using Right Endpoints The right endpoints of the subintervals are: x1 = a + ∆x, x2 = a + 2∆x, x3 = a + 3∆x, . Then the area of the ith rectangle is f(xi)∆x. So the area of S Is approximated by the sum of the areas of these rectangles: Rn = f(x1)∆x + f(x2)∆x + … + f(xn)∆x

More Generally

More Generally (cont’d) Since this approximation seems to become better and better as n increases, we define the area A of the region S as follows: Definition The area 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: Sigma Notation:

Example 2 A particle starts at x = 0 and moves along the x-axis with velocity v(t) = 5 for time x ≥ 0. Where is the particle at t = 4? A particle starts at x = 0 and moves along the x – axis with velocity v(t) = t2 +1 for time x ≥ 0. Where is the particle at t = 4? Approximate the area under the curve using four rectangles.

Example 3 Let A be the area of the region that lies under the graph of f(x) = e-x between x = 0 and x = 2 Using right endpoints, find an expression for A as a limit. Do not evaluate the limit. Estimate the area by taking the sample points to be midpoints and using four subintervals.

The Distance Problem Here our goal is to find the distance traveled by an object during a certain time period if the velocity is known at all times. If the velocity is constant, then the problem reduces to the familiar formula distance = velocity * time But what if the velocity varies?

Example 4 Suppose the odometer on our car is broken and we want to estimate the distance driven over a 30-second time interval. Time (s) 5 10 15 20 25 30 Velocity (ft/s) 31 35 43 47 46 41

Connection with Area On the next slide we Sketch the velocity function of the car, and Draw rectangles whose heights are the initial velocities for each time interval.