Set up and evaluate the definite integral that gives the area of the region bounded by the graph

Set up and evaluate the definite integral that gives the...

Key Concepts

Differentiation for Slope Calculation

Differentiation is used to compute the slope of the tangent line to the curve at a given point by calculating the derivative of the function. This slope is a critical piece of information for constructing the equation of the tangent line.

Definite Integral for Area Between Curves

This concept involves computing the area of a region bounded by two curves by integrating the difference of the functions over a specified interval. The method requires identifying the upper and lower functions within the interval and setting up an integral with limits corresponding to their points of intersection.

Tangent Line to a Curve

The tangent line at a specific point on a curve represents the best linear approximation of the function at that point. Determining the tangent line involves computing the derivative of the function at the point and using point-slope form to write the line’s equation.

Finding Intersection Points

To determine the limits of integration, one must solve for the points where the curves intersect. This usually entails setting the function and the tangent line equal to each other and solving the resulting equation, which provides the boundaries for the definite integral.

Transcript

00:01 There's a lot going on in this problem, so we'll just jump right into it.

00:04 I would expect my students to know x cubed is a function that looks like this.

00:11 F of x equals x cubed.

00:13 The first thing to notice is that they ask you at the point 1 -1 what the tangent line is.

00:22 Well, if you did the derivative, well, f prime of x, we know how to find the derivative by moving the exponent in front, subtracting 1 from that exponent.

00:33 And when you plug in 1 for the derivative, 1 squared is just 1 times 3 is 3.

00:40 So this is the slope of the tangeline.

00:44 So i would just kind of lazily just think about how the slope is 3 and how to get to up 3 over 1 to find the y intercept.

00:54 I would have to go down 3 then.

00:57 I do not do a very good job.

00:59 But the point i'm trying to make is the lower function would be y equals 3x minus 2.

01:07 So when we find the area, and again, i kind of made this not look very good, we already know that the upper bound will be, well, it won't intersect again to the right.

01:20 So that's why i know that's going to be the integral from something to one of the upper function, which is x cubed minus that lower function of 3.

01:30 X minus 2 so don't forget to distribute that minus to that negative 2 as well d x now how do i find that upper bound uh i'll have to set or sorry the lower bound uh x cubed equal to 3x minus 2 um some people might need to use a calculator it's got to be a negative number so you could just guess and check like negative 1 cubed and see if that's going to equal 3 times negative 1 is negative 3 minus 2...

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