Find the area enclosed by the curves y=c-x^2 and y=x^2-c as a function of c . Find the value of

Find the area enclosed by the curves y=c-x^2 and y=x^2-c as...

Key Concepts

Finding Intersection Points

To determine the region enclosed by curves, it is essential to find the points at which the curves intersect. This usually involves equating the two functions and solving the resulting equation. The intersection points serve as the limits of integration, and understanding this concept is vital in setting up the correct integral to compute the area.

Solving for a Parameter

In problems where the area has to satisfy a condition, such as equating the computed area to a given value, it becomes necessary to solve for a parameter within the functions. This involves forming an equation with the parameter and using algebraic or numerical methods to determine its value, which is a common technique in both pure and applied calculus contexts.

Area Between Curves

This concept involves calculating the region between two functions by integrating the difference of the functions over the appropriate interval. It is a fundamental application of definite integrals, where one must identify the upper and lower functions across the range of integration. The concept is key in many applications of calculus where areas under or between curves need to be determined.

Setting up Definite Integrals

When calculating the area between curves, one sets up a definite integral by first determining the limits of integration, which are usually found by solving for the points where the curves intersect. This process involves subtracting the lower function from the upper function over the interval defined by these limits, and then evaluating the resulting integral.

Transcript

00:01 Find the area enclosed by y equals c minus x squared and y equals x squared minus c and then find what value of c makes this area equal to one.

00:11 All right, let's have a look at this.

00:13 Y equals c minus x squared, that is a parabola, and that parabola is upside down and scooted up c.

00:25 All right, so here it will be.

00:30 All right.

00:30 And then y equals x squared minus c, that's a parabola.

00:34 It's right side up and scooted down c.

00:37 So here it will be.

00:44 Okay, to find the area, we need to cut this into slices.

00:48 And what we want to do on this one is cut the slices up and down.

00:52 And you can see that that's the right thing to do because no matter where i put the slice here, here, here, the red one is on the top and the blue one is on the bottom.

01:04 Okay, if we cut it the other way, some of them will have red on both sides and some of them will have blue on both sides.

01:09 So not only would we have to solve the equations for x, but we have to do two integrals.

01:15 So you always try and make it so that you have the same function on the top and the same function on the bottom for no matter what slice you pick.

01:25 All right, so we need the height.

01:27 So the height will be y on the top minus y on the bottom...

Please give Ace some feedback