Use a definite integral to find the area of the region between the given curve and the x -axis o

Use a definite integral to find the area of the region...

Key Concepts

Definite Integral

A definite integral represents the accumulation of quantities, such as area under a curve, over a closed interval. It is computed by evaluating the limit of Riemann sums and incorporates the limits of integration to provide a numerical value for the accumulated quantity.

Area Under a Curve

The area under a curve is calculated using a definite integral, which sums infinitely many infinitesimal vertical slices bounded by the curve and the x-axis over a specified interval. This approach is fundamental in finding the total area enclosed by a curve and an axis.

Power Rule for Integration

The power rule for integration is a basic technique used to find antiderivatives of functions in the form x^n, where n is any real number except -1. It simplifies the process of finding the integral by increasing the exponent by one and dividing by the new exponent.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus bridges the concepts of differentiation and integration by asserting that the definite integral of a function over an interval can be determined using any one of its antiderivatives. This theorem provides an efficient method to calculate areas and other accumulated quantities.

Transcript

00:01 What we want to do is we want to use a definite integral to find the area, which is bounded by the equation y equal to 3x squared and the x -axis from the interval from 0 to b.

00:35 Okay, and so we know that is the integral from 0 to b of 3x squared dx.

00:51 Okay, and so what we want to do is we also know that that definite integral is related to the limit as the partitions of those rectangles goes to 0 of the sum as k equals 1 to n of 3 c sub k squared times delta x sub k.

01:26 Okay.

01:27 And so now we need to kind of define that delta x sub k and that c sub k.

01:34 And so we know that delta x of k is equal to b minus a over in or the number of partitions.

01:49 And so this is just going to be b over in because a is zero.

01:55 Now we're going to be using the right end point.

02:03 And so c of k is equal to a plus k delta x of k, and so this is just going to be equal to a delta x of k, which is going to be equal to k times b over n.

02:29 Okay, and so what we're going to do is substitute all those into the limit, and so this is going to be another right way to write the limit as the partition goes to zero is that the limit as the number of rectangles goes to infinity of the sum of k equal one to n of three times kb over n and we're going to square that times b over n.

03:04 Okay and so now we're going to kind of simplify a little bit now anything that doesn't have a in in it can be pulled out to the very very front of the limit and so the three is going to be pulled out the b cubed and we have a b cubed because we have a b squared times a b that can be pulled out to the very front and then what we have is the limit as n goes to infinity.

03:42 Now i have 1 over n cubed can be pulled out of that summation...

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