Just diving into integral calculus

[1MileCrash]

And the first thing I'm tackling is understanding the area of a shape by inscribed/circumscribed polygons.

For example, the shape produced by the line: y = x + 1, the y axis, and the vertical line x = 2.

Now, my understanding is that the area is the limit as n approaches infinity of the sum of the area of n inscribed rectangles.

Can someone explain how this translates to actual math? How would I find the area of the above shape?

 

[Mark44]

Is the other boundary of your region the x axis? If that's the region, its shape is a trapezoid whose area is 4.

Using the limit of a Riemann sum to get the value, you would divide the interval [0, 2] into some number of subintervals, and the use rectangles or some other shape to approximate the area above each subinterval.

For example, if the interval is divided into 4 subintervals, and inscribed rectangles are used, the area estimate is found to be .5*1 + .5*1.5 + .5 * 2.0 + .5 * 2.5 = .5*(7) = 3.5.

To get a better approximation, divide the interval into more subintervals.

 

[1MileCrash]

Is the other boundary of your region the x axis? If that's the region, its shape is a trapezoid whose area is 4.

Is it a valid method for me to do F(2) - F(0) to arrive at that answer? Would I be applying the theorem correctly?

$F(x) = \frac{x^2}{2} + x$

F(2) = 4
F(0) = 0

4-0 = 4

Since we are bounded by x = 0 and x = 2?

 

[dalcde]

Yes.

 

[1MileCrash]

That's really powerful!