Integrals / Riemann Sums

Question

Estimate the area under the graph of the function f(x)=x+5−−−−√ from x=−3 to x=3 using a Riemann sum with  n=10 subintervals and midpoints.

 

Round your answer to four decimal places.

 

area =  ____________ 

Answer 1

Question: Estimate the area under the graph of the function y = sqrt(x+5) from x = -3 to x = 3 using a Riemann sum with n = 10 sub-intervals and midpoints.

Answer: Before proceeding, I don't know what happened with the formatting of the function y, so I'm interpreting it to mean y = sqrt(x+5).

Now, what is the distance between x = -3 and x = 3 ? It is 3 - (-3) = 6. We need ten subdivisions, so each rectangle we'll be using is (6)/(10) = 0.6 units wide. (This is the Δx = (b-a)/n formula in most textbooks.)

Now we know how wide our rectangles are. How how high are they? That changes over the course of the function. Let's get to work finding midpoints:

We need to cover the ground from x = -3 to x = 3. We'll divide this up into ten sub-intervals of 0.6 units in length. We'll get:

(-3), -2.4, -1.8, -1.2, -0.6, 0, 0.6, 1.2, 1.8, 2.4, 3.

Excluding that first (-3), what we have are actually right-endpoints for each rectangle. We need midpoints, not right-hand boundaries. Let's go in between these numbers (half of 0.6 is 0.3):

-2.7, -2.1, -1.5, -0.9, -0.3, 0.3, 0.9, 1.5, 2.1, 2.7.

These are our midpoints. Evaluate y for each of these points:

-2.7: sqrt(-2.7 + 5) is approximately 1.516575

-2.1: sqrt(-2.1 + 5) is approximately 1.702939

-1.5: sqrt(-1.5 + 5) is approximately 1.870829

-0.9: sqrt(-0.9 + 5) is approximately 2.024846

-0.3: sqrt(-0.3 + 5) is approximately 2.167949

0.3: sqrt(0.3 + 5) is approximately 2.302173

0.9: sqrt(0.9 + 5) is approximately 2.428992

1.5: sqrt(1.5 + 5) is approximately 2.549510

2.1: sqrt(2.1 + 5) is approximately 2.664583

2.7: sqrt(2.7 + 5) is approximately 2.774887

Add those values together: 22.003283.

Multiply this sum by the Δx value we found earlier, 0.6.

0.6 * 22.003283 ≈ 13.2020

The exact value of the integral shows on my calculator as 13.19932658, so we did well here, with an error of 0.0027.

The area under the graph on the specified interval is estimated to be 13.2020.