Calculus- Integration (Area/Sums)?

Question

What is the integral from '1' to '3' of 'e^(x^2)dx? You must find it using the right endpionts, and 5 rectangles.

I know the formula for right endpoints is the sum of R=1 to 'n' of 'f(XsubR)(change in x)

change in x= (b-a)/n... n is the number if rectangles

Answer 1

Hello,
Lets first break down the question, we have some function that we want the area of underneath it using 5 rectangle within an x domain of 1 to 3. So we know that the change in x is 2, and that 2/5 is the increment of x values starting from (1+2/5) which is the x value to the right end of our first rectangle, and is also equal to (1.4). Ok so to get the y value we do: f(1.4), so we have the x and y value for the first right end rectangle, multiply the two to get area of the first rectangle, ok so lets sum this whole thing up:
Area = x*y] for all x,y in range and domain
which translates to:
Area = x1y1 + x2y2 + x3y3 + x4y4 + x5y5
We already know all of the x values because the first rectangle's right edge is at: 1.4 and each next rectangle has an x value 0.4 greater than the last so lets fix the equation:
Area = (0.4)y1 + (0.4)y2 + (0.4)y3 + (0.4)y4 + (0.4)y5
Ok so that checks out because remember every rectangle has an equal width or change in x value and it is 0.4. Now for the y values:
A = (0.4)(f(1.4)) + (0.4)(f(1.8)) + (0.4)(f(2.2)) + (0.4)(f(2.6)) + (0.4)(f(3.0))
ok so now we know all that we need just to find the y values we need to plug in each x value of the function y = f(x) to replace with a number, for example y(1.4) = e^(1.4^2), and so on for each f(x) value. So lets rewrite:
Area = (0.4)e^(1.4^2) + (0.4)e^(1.8^2) + (0.4)e^(2.2^2) + (0.4)e^(2.6^2) + (0.4)e^(3.0^2)
Thats all, we can leave it like this or you can use a calculator to solve for the exact value. Make sure if you use a calculator you are careful with the parenthesis involved. Also notice how the f(x) values are different for each one, this is because the function f(x) increases as x increases so each value of x had to be entered in order to find the appropriate height at that rectangle edge, also known as the y value.

Answer 2

Hello,

We have delta x = (3-1)/5 = 2/5 so the first right hand end point is 1 + 2/5 or 7/5 then the other four are 9/5, 11/5, 13/5, 3.

now our equation is

Area = 2/5 *e^(49/25) + 2/5 * e^(81/25) + 2/5* e^(121/25) + 2/5 * e^(169/25) + 2/5 e^3

I'll let you evaluate it.

Hope This Helps!!