Approximate the area under the curve of f(x)=9-x^2 between x=-2 and x=3 by calculating the area of 5 rectangles for both left hand endpoints and right hand endpoints.
I have the left hand and right hand endpoints, what I need help in is solving for the actual area.
Thanks
Remember that the area is the sum of little rectangles, with width 1 and height f(x) at the endpoints.
There are lots of good online Riemann sum calculators, such as this one:
http://www.emathhelp.net/calculators/calculus-2/riemann-sum-calculator/
Just plug in your data and see the calculations to verify your results.
To approximate the area under the curve, you can use the method of left and right Riemann sums. Riemann sums divide the interval between x=-2 and x=3 into smaller subintervals and approximate the area under the curve as the sum of the areas of rectangles.
Let's start by calculating the width of each rectangle. Since you are using 5 rectangles, the total width of the interval is (3 - (-2)) = 5. Therefore, each rectangle has a width of 5/5 = 1.
Next, we need to calculate the height of each rectangle. For the left hand Riemann sum, the height of each rectangle is the value of the function at the left hand endpoint of each subinterval. Given f(x) = 9 - x^2, we can plug in the left hand endpoints of the subintervals to find the height for each rectangle.
For the given interval, the left hand endpoints are -2, -1, 0, 1, and 2. Plugging these values into the function, we get:
f(-2) = 9 - (-2)^2 = 9 - 4 = 5
f(-1) = 9 - (-1)^2 = 9 - 1 = 8
f(0) = 9 - (0)^2 = 9
f(1) = 9 - (1)^2 = 9 - 1 = 8
f(2) = 9 - (2)^2 = 9 - 4 = 5
Now let's calculate the area of each rectangle for the left hand endpoints. The area of a rectangle is given by the product of its width and height.
The width is 1 for each rectangle. The heights we just calculated are 5, 8, 9, 8, and 5. Therefore, the areas of the rectangles are:
Rectangle 1: 1 * 5 = 5
Rectangle 2: 1 * 8 = 8
Rectangle 3: 1 * 9 = 9
Rectangle 4: 1 * 8 = 8
Rectangle 5: 1 * 5 = 5
Finally, to calculate the approximate area under the curve using left hand endpoints, we sum up the areas of all the rectangles:
Total area ≈ 5 + 8 + 9 + 8 + 5 = 35
You can repeat the same process for the right hand endpoints, where the height of each rectangle would be calculated using the right hand endpoint of each subinterval. The rest of the calculations would be the same.
I hope this explanation helps! Let me know if you have any further questions.