Section Area:
Use Riemann sums and a limit to compute the exact area under the curve.
y = 4x^2 - x on (a) [-0,1]; (b) [-1, 1]; (c) [1, 3]
To compute the exact area under the curve using Riemann sums and a limit, here are the step-by-step instructions for each interval:
(a) Interval [-0,1]:
1. Divide the interval [-0,1] into smaller subintervals. The smaller the subintervals, the more accurate the approximation of the area.
2. Choose a sample point within each subinterval. This sample point will be used to evaluate the function.
3. Calculate the width of each subinterval.
4. Evaluate the function, y = 4x^2 - x, at each sample point.
5. Multiply the function value by the width of the corresponding subinterval.
6. Sum up all the individual products from step 5 to find the approximate area under the curve.
(b) Interval [-1, 1]:
Follow the same steps as in interval (a), but adjust the lower and upper bounds of the interval, and recalculate the width of each subinterval accordingly.
(c) Interval [1, 3]:
Again, repeat the steps from interval (a), but adjust the lower and upper bounds of the interval and calculate the width of each subinterval based on the new values.
After following these steps, you will have an approximation of the area under the curve for each interval. To compute the exact area, you can take the limit as the width of the subintervals approaches zero. This means that you will repeat the steps with an increasing number of subintervals and use a smaller width for each subdivision. The more subintervals you use, the more accurate your approximation will be. Finally, by taking the limit as the width approaches zero, you will arrive at the exact area under the curve for each interval.