Question

The graph of a function f is given. Estimate ∫010​f(x)dx using five subintervals with the following. (a) right endpoints (b) left endpoints (c) midpoints

Answer 1

To estimate the integral of a function using five subintervals, one can use the right endpoints, left endpoints, or midpoints method and sum up the areas of the rectangles formed by these values and the subinterval width. Without the exact graph shape, geometric area calculations for shapes like horizontal lines or right triangles could be used based on the described function.

Step-by-step explanation:

To estimate the integral ∫010​f(x)dx using five subintervals, we apply different methods such as right endpoints, left endpoints, and midpoints. Let's break down the integral into subintervals of width 2 since 10 divided by 5 equals 2. Here's a step-by-step explanation for each method:

• Right endpoints: We take the value of the function f(x) at the right end of each subinterval and multiply it by the width (2) to calculate the area of each rectangle. Then we sum all these areas to get the estimate of the integral.
• Left endpoints: Similarly, we consider the value of function f(x) at the left end of each subinterval, multiply by the subinterval width, and again sum all the areas.
• Midpoints: For this method, we take the value of f(x) at the middle point of each subinterval. This usually gives a better approximation than the other two methods when the function is fairly smooth.

Since the exact shape of the graph is not provided, we cannot calculate the exact areas. However, if the function were described as a right triangle or a horizontal line, the area under the curve would correspond to the geometric area of the shape. For a horizontal line, it would be the product of the function's value and the interval length, while for a right triangle it would involve 1/2 * base * height calculation.

Answer 2

To estimate the definite integral, we can use right endpoints, left endpoints, or midpoints of the subintervals.

Step-by-step explanation:

To estimate ∫₀¹₀​f(x)dx using the right endpoints, we divide the interval [0,10] into five subintervals of equal width. The right endpoints of these subintervals are 2.5, 3, 3.5, 4, and 4.5. We then evaluate f(x) at these right endpoints and sum up the products of the function values and the width of each subinterval. This will give us an approximation of the definite integral.

To estimate ∫₀¹₀​f(x)dx using the left endpoints, we follow the same steps as above. However, this time we use the left endpoints of the subintervals, which are 2, 2.5, 3, 3.5, and 4.

To estimate ∫₀¹₀​f(x)dx using the midpoints, we find the midpoint of each subinterval. The midpoints are 2.25, 2.75, 3.25, 3.75, and 4.25. We then evaluate f(x) at these midpoints and sum up the products of the function values and the width of each subinterval.