Approximate the area under each curve by evaluating te function at the left-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.
You would have four trapezoids in your approximation.
the width of each of those traps is .5
and the heights at
x = 1, 1.5, 2, 2.5, and 3 are
h = 8, 6.75, 5, 2.75, and 0 respectively
the area is appr. = .5/2[(8+6.75)+(6.75+5)+(5+2.75)+(2.75+0)]
= 9.25
(the exact answer would have been 9.33333..)
do the second one the same way.
To approximate the area under each curve by evaluating the function at the left-hand endpoints of the subintervals, follow these steps:
1. Divide the given interval into the desired number of subintervals. For example, in question 1, the interval is from x=1 to x=3, and we want to use 4 subintervals. Therefore, each subinterval would have a width of (3-1)/4 = 1/2.
2. Determine the left-hand endpoints of each subinterval. In this case, we start at x=1 and move along the interval in increments of 1/2 to find the left-hand endpoints of each subinterval. The left-hand endpoints would be 1, 1.5, 2, and 2.5.
3. Evaluate the function at each left-hand endpoint to find the corresponding function values. For question 1, the function is f(x) = 9 - x^2. So, we'll evaluate this function at each of the left-hand endpoints found in step 2. The function values would be:
- f(1) = 9 - 1^2 = 8
- f(1.5) = 9 - 1.5^2 = 6.75
- f(2) = 9 - 2^2 = 5
- f(2.5) = 9 - 2.5^2 = 3.75
4. Calculate the area of each rectangle by multiplying the width of each subinterval by its corresponding function value. In this case, the width of each subinterval is 1/2. So, we'll multiply 1/2 by each of the function values obtained in step 3 to get the areas:
- Area 1 = (1/2) * 8 = 4
- Area 2 = (1/2) * 6.75 = 3.375
- Area 3 = (1/2) * 5 = 2.5
- Area 4 = (1/2) * 3.75 = 1.875
Now, you have the approximate areas under the curve for each subinterval evaluated at the left-hand endpoints.