I'm a little confused with this integration problem: If the definite integral from 0 to 2 of (e^(x^2)) is first approximated by using two inscribed rectangles of equal width and then approximated by using the trapezoidal rule with n=2, the difference between the two approximations is what?

The answer is 26.80

Steve's answer is unfortunately incorrect. 30.517 was wrong when picked.

Steve is correct. He gave us both the LH & RH approximations as well as the trapezoid approximation

You just needed to subtract Trapezoid & the LH which gives 30.517-3.718 = 26.80

Thank you!

Ah, integration problems can be a bit confusing, but fear not, for I am here to help!

So, you want to find the difference between the approximation using two inscribed rectangles and the trapezoidal rule with n=2, right? Well, let's dive in!

First, let's approximate the integral using two inscribed rectangles. When you use rectangles, you either overestimate or underestimate the actual area, right? In this case, the rectangles are inscribed, which means they're underestimating the area. So, our approximation using two inscribed rectangles will be an underestimate.

Now, let's move on to the trapezoidal rule with n=2. The trapezoidal rule is a fancy name for approximating the integral using trapezoids. And we all know that trapezoids are like the cool kids of shapes, right?

When we use the trapezoidal rule, we approximate the area under the curve by connecting the endpoints with straight lines and creating a series of trapezoids. In this case, with n=2, we'll have 2 trapezoids.

Now, the trapezoidal rule tends to be more accurate than the rectangle method. Why? Because trapezoids are more versatile and can fit the curve better, just like a good pair of jeans that hugs your curves in all the right places!

So, to answer your question, the difference between the two approximations will be the difference between an underestimate and a more accurate approximation. The trap-easy-does-it trapezoidal rule will give us a better approximation than the rectangles, which means the difference will be positive!

I hope that clears things up for you! If you have any more questions, feel free to ask.

To find the difference between the approximations obtained using inscribed rectangles and the trapezoidal rule, we first need to calculate the approximations separately and then subtract one from the other.

Let's start by approximating the definite integral using inscribed rectangles of equal width. In this case, we divide the interval [0, 2] into two subintervals of equal width. The width of each subinterval is given by Δx = (2 - 0) / 2 = 1.

For each subinterval, we choose the left endpoint as the height of the rectangle. Since we have two subintervals, we need to evaluate the function e^(x^2) at x = 0 and x = 1.

Approximation using inscribed rectangles:
f(0) = e^(0^2) = e^0 = 1
f(1) = e^(1^2) = e^1 = e

Now, we calculate the area of each rectangle as the height (function value) multiplied by the width (Δx) and sum them up:

Area of the first rectangle = 1 * 1 = 1
Area of the second rectangle = e * 1 = e

Total approximation using inscribed rectangles = Area of the first rectangle + Area of the second rectangle = 1 + e

Next, let's approximate the definite integral using the trapezoidal rule with n = 2. The trapezoidal rule approximates the area under the curve by approximating the curve with a series of trapezoids.

For n = 2, we divide the interval [0, 2] into two subintervals of equal width. Again, the width of each subinterval is given by Δx = (2 - 0) / 2 = 1.

We need to evaluate the function e^(x^2) at x = 0, x = 1, and x = 2.

Approximation using the trapezoidal rule:
f(0) = 1 (same as before)
f(1) = e (same as before)
f(2) = e^(2^2) = e^4

Now, we calculate the area of each trapezoid using the formula: (f(a) + f(b)) * Δx / 2, where a and b are the endpoints of each subinterval.

Area of the first trapezoid = (1 + e) * 1 / 2 = (1 + e) / 2
Area of the second trapezoid = (e + e^4) * 1 / 2 = (e + e^4) / 2

Total approximation using the trapezoidal rule = Area of the first trapezoid + Area of the second trapezoid = (1 + e) / 2 + (e + e^4) / 2 = (e^4 + 3 * e + 1) / 2

To find the difference between the two approximations, we subtract the approximation using inscribed rectangles from the approximation using the trapezoidal rule:

Difference = Total approximation using the trapezoidal rule - Total approximation using inscribed rectangles
= (e^4 + 3 * e + 1) / 2 - (1 + e)

Simplifying this expression further will give you the numerical difference between the two approximations.

x f(x)

0 1
1 e
2 e^4

So, if there are 2 rectangles of width 1, then the area, using left-sides is

1*1 + 1*e = e+1 = 3.718

using right-sides, it's

1*e + 1*e^4 = 57.316

Using the trapezoidal rule, we have

1(1+e)/2 + 1(e+e^4)/2 = 30.517

Kind of a coarse approximation.