Question

Let f(x) = 4x. (a) Sketch the region R under the graph of f on the interval [0, 2]. Find its exact area using geometry. square units (b) Use a Riemann sum with four subintervals of equal length (n = 4) to approximate the area of R. Choose the representative points to be the left endpoints of the subintervals. square units (c) Repeat part (b) with eight subintervals of equal length (n = 8). square units (d) Compare the approximations obtained in parts (b) and (c) with the exact area found in part (a). Do the approximations improve with larger n? Yes / No

          Let  f(x)
= 4x.
(a) Sketch the region R under the
graph of f on the interval 
[0, 2].
 Find its exact area using geometry.
                 
    square units
(b) Use a Riemann sum with four subintervals of equal
length (n = 4) to approximate the area
of R. Choose the representative points to be the left
endpoints of the subintervals.
                 
         square units
(c) Repeat part (b) with eight subintervals of equal
length  (n = 8).
                 
             square units
(d) Compare the approximations obtained in parts (b) and
(c) with the exact area found in part (a). Do the approximations
improve with larger n?
Yes / No

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Added by Patrick M.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Kathleen Carty

12/07/2022

Step 1

The region R under the graph of f on the interval [0, 2] is a triangle with base 2 and height 4*2 = 8. The area of a triangle is 1/2 * base * height, so the exact area of R is 1/2 * 2 * 8 = 8 square units. Show more…

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Let f(x) = 4x. (a) Sketch the region R under the graph of f on the interval [0, 2]. Find its exact area using geometry. square units (b) Use a Riemann sum with four subintervals of equal length (n = 4) to approximate the area of R. Choose the representative points to be the left endpoints of the subintervals. square units (c) Repeat part (b) with eight subintervals of equal length (n = 8). square units (d) Compare the approximations obtained in parts (b) and (c) with the exact area found in part (a). Do the approximations improve with larger n? Yes / No

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Transcript

00:01 So i want to graph the function y equals 4 times x on the interval 0 to 2.

00:10 So on the interval 0 to 2, at 0 the function is 2, at 2 the function is 8.

00:20 So that's going to look like this.

00:22 And if i want to find that region under the curve, then i'm just going to drop this segment.

00:29 So what i'm looking for is this area, right? that exact area.

00:36 So that is a triangle, and the area is half base times height.

00:43 So that base and height happen right here where the right angle forms.

00:49 So half, my base is two.

00:51 Now i need my height.

00:53 My height is the value of the function when i put into, so my height is eight.

00:58 So that is going to give me half of two is one.

01:02 That's going to be eight square units.

01:06 So that's my area using geometry.

01:11 Now i want to use ramon's sum with four sub -intervals.

01:16 So 2 minus 0 divided by 4 means that each of my sub -intervals is going to be a half in length.

01:25 So if you think about that, you're starting at 0, you're going out.

01:30 To two and you want four of these.

01:32 So you want 0 .5, 1, and 1 .5.

01:39 Now we want to use a left endpoint.

01:44 So we want 0 .5, 1, and 1 .5.

01:49 F of 0 is 0, f of 0, f of 0, 5 is 2, f of 1 .5 is 2, f of 1 is 4, and f of 1 .5 is six...

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