Use a graph to give a rough estimate of the area of the...

Key Concepts

Definite Integration

Definite integration is the process in calculus of calculating the accumulation of quantities, which is typically used to determine the area under a curve between specified limits. It involves computing the integral of a function over a closed interval, and the result represents the net area under the function’s graph relative to the axis over that interval.

Area Under a Curve

The area under a curve is a key concept in calculus that refers to the region bounded by a curve, the x-axis, and vertical lines at the endpoints of an interval. This area can be precisely determined using definite integrals, and it represents the total accumulation or net value of the function across the interval.

Graphical Estimation of Area

Graphical estimation of area involves analyzing the graph of a function to visually approximate the area under the curve. This technique provides an intuitive understanding of integration by estimating the region’s size using geometric shapes or numerical methods before performing the exact calculation through integration.

Substitution Method

The substitution method is a technique used in integration to simplify complex integrals by changing variables. When the integrand includes a function of a linear expression, a substitution can transform the integral into a more familiar form, making it easier to evaluate. This method is particularly useful in handling integrals involving composite functions like square roots of linear expressions.

Transcript

00:01 Okay, so we have the function, y equals 2x plus 1 to the 1 half power, and on the range from 0 to 1, or the domain from 0 to 1.

00:11 So what we want to do is first estimate the area using its graph and then proceeding to find the exact area by integrating it.

00:21 So in order to estimate this area, we're going to use an estimation technique called the trapezoidal rule, where we have two points a and b.

00:29 And we're going to use those to form a trapezoid that is made underneath the graph.

00:37 So the region under the graph is going to make me look like a trapezoid, and we're going to find the area of that trapezoid that we make, and that's how we can estimate a definite integral.

00:46 So we go ahead and kind of finish drawing out this trapezoid.

00:52 We can see that it's pretty close to what our graph looks like, or what this region looks like underneath the red line here.

00:59 So we're going to use that as estimation technique to approximate this.

01:02 This area.

01:04 And here's the equation.

01:05 So to estimate this integral, we're going to essentially take the area of this trapezoid we just made, where b is the point, is the point one, and its y value, and point a is zero, and its y value.

01:19 We can find its y value by just plugging in to our expression for y and go from there.

01:25 That's what we're going to do.

01:27 So we get this, so equals to b minus a, which is just 1 minus 0.

01:33 Those are the x points.

01:36 And times one half an f of a is just when we plug in zero into our y function that's just one we have to plug in for f of b so we're going to do two times one plus one and take the square root of that so that's going to be the square root of three square root of three there we go and we can simplify that a bit so our estimate for this area using the trapezoid or rule is just one half times one plus root three.

02:24 So there's our estimate.

02:26 It's not going to be exact, but again, it's something we can work with moving into doing the actual integral to get the exact area.

02:36 So there's our approximation using the troposota rule.

02:39 We'll keep going down here...

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