Approximate the area under each curve over the specified interval by using the indicated number of subintervals and evaluating the function at the right-hand endpoints of the subintervals.
1. f(x)=9-x^2 from x=1 to x=3; 4 subintervals.
2. f(x)=x^2 +x+ from x= -1 to x=1; 4 subintervals.
To approximate the area under each curve using the right-hand endpoints of the subintervals, you can use the Riemann sum method. This involves dividing the interval into subintervals and then evaluating the function at the right-hand endpoint of each subinterval.
1. f(x) = 9 - x^2 from x = 1 to x = 3; 4 subintervals:
- First, determine the width of each subinterval by dividing the total interval width by the number of subintervals. In this case, the interval width is 3 - 1 = 2, so each subinterval has a width of 2/4 = 0.5.
- Next, determine the right-hand endpoint for each subinterval. The right-hand endpoints are given by adding the subinterval width to the left endpoint of each subinterval. So for the first subinterval, the right-hand endpoint is 1 + 0.5 = 1.5. For the second subinterval, the right-hand endpoint is 1.5 + 0.5 = 2, and so on.
- Finally, evaluate the function at each right-hand endpoint and multiply the function value by the subinterval width. Then, sum up all the results to approximate the area under the curve.
2. f(x) = x^2 + x from x = -1 to x = 1; 4 subintervals:
- Again, find the interval width by subtracting the left endpoint from the right endpoint: 1 - (-1) = 2. So each subinterval has a width of 2/4 = 0.5.
- Determine the right-hand endpoint for each subinterval by adding 0.5 to the left endpoint. For the first subinterval, the right-hand endpoint is -1 + 0.5 = -0.5. For the second subinterval, the right-hand endpoint is -0.5 + 0.5 = 0, and so on.
- Evaluate the function at each right-hand endpoint, multiply the function value by the subinterval width, and sum up the results to approximate the area under the curve.
Remember that these approximations are not exact solutions but provide an estimation of the area under the curve.