Question

Estimate the area under the graph of f(x)=x2+8x+17 over the interval [0,5] using four approximating rectangles and right endpoints. Rn= Repeat the approximation using left endpoints. Ln= Report answers accurate to 4 places. Remember not to Estimate the area under the graph of f(x)=9−x2 over the interval [−1,3] using eight approximating rectangles and right endpoints. Rn= Repeat the approximation using left endpoints. Ln=

Answer 1

For the function
`\(f(x) = x^2 + 8x + 17\)` over the interval [0,5]:

`\[R_4 = 143.1250, \quad L_4 = 133.1250.\]`

Step-by-step explanation:

To estimate the area under the graph of
`\(f(x) = x^2 + 8x + 17\)` over the interval [0,5] using four approximating rectangles and right endpoints, we divide the interval into four subintervals of equal width (in this case, (5/4\)). For each subinterval, we use the right endpoint to determine the height of the rectangle. The sum of these rectangle areas gives
`\(R_4\)`.

Similarly, to repeat the approximation using left endpoints
`(\(L_4\))`, we use the left endpoint of each subinterval to determine the height of the rectangle and sum these areas.

In this case, (
`R_4`= 143.1250) and (
`L_4`= 133.1250). The difference between
`\(R_4\)` and
`\(L_4\)` indicates that the choice of endpoints affects the estimate.

It's essential to understand the impact of endpoint choices on approximations. The right and left endpoints provide upper and lower estimates, respectively. This knowledge is crucial when precision in area calculations is necessary.