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 is a numerical calculation of the accumulation of the quantity under a curve between two limits. It is typically represented with an integral sign containing a lower limit and an upper limit and is evaluated using antiderivatives.

Area Under the Curve

The area under a curve is computed by integrating the function defining the curve between two points. This process measures the accumulated 'area' between the function and the x-axis over a specific interval.

Antiderivative

An antiderivative of a function is another function whose derivative matches the original function. When used in definite integrals, it allows the application of the Fundamental Theorem of Calculus to evaluate areas.

Limits of Integration

Limits of integration are the values that define the interval over which the integration is performed. They indicate the starting and ending points on the x-axis where the area under the curve is computed.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus links the concept of differentiation with that of integration by stating that the definite integral of a function can be calculated using its antiderivative evaluated at the boundary values given by the limits of integration.

Transcript

00:01 Okay, what we want to do is we want to use a definite integral to find the area.

00:13 And the area is bounded between the graph of y equal to x over 2 plus 1 and the x axis on the interval from 0 to b.

00:25 And so we know this is the integral from 0 to b of x over 2 plus 1 dx, which any time we have addition.

00:35 Of subtraction, we're going to break them into two separate.

00:38 So i have a one -half times the integral from 0 to b of x dx, plus the integral from 0 to 1 of 0 to b of just 1 times dx.

00:51 Okay, and we know that these can be replaced with the limit.

00:57 So it's 1 half.

00:59 The limit as the partition of the rectangles goes to 0.

01:03 Of the sum, k equal to 1 to n of c sub k delta x of k plus the limit as the partitions go to 0 of the sum k equal 1 to n of here i just have a d x so this is just going to be a delta x sub k.

01:35 Okay, so now we need to define what delta x of k is and cc is.

01:43 So delta x of k is the width of each of those rectangles.

01:49 So typically it's defined to be b minus a over n.

01:53 So in our case it's just going to be b over in.

01:57 And c of k is the right endpoint or the right in point height of each of those rectangles, which is going to be a plus k times delta x sub k.

02:13 So this is going to be kb over n.

02:16 Okay.

02:17 So these limits now become the one half, the limit as n goes to infinity this time.

02:25 So i've changed those on you.

02:27 So saying that the partitions go to zero, it's the same thing as saying the number of the rectangles increases...