Cal III - Area Between the Parametric Curve Help with Finding the Bounds

[TChao]

Find the area between the parametric curve, x(t)=cos(t), y(t)=sin^2(t) and the x-axis

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My work shown in the link provided above without the bounds founded. Sorry for not rotating the image and poor quality. I need help trying to understand how to find the bounds for this equation.

 

[HallsofIvy]

Surely, you know that \(\displaystyle sin^2(t)+ cos^2(t)= 1\)? So \(\displaystyle x^2+ y= cos^2(t)+ sin^2(t)= 1\). \(\displaystyle y= 1- x^2\) is a parabola with vertex at (0, 1) opening downward. It crosses the x-axis (y= 0) at x= -1 and x= 1.

You could also have observed that \(\displaystyle y= cos^2(t)= 0\) when t is an odd multiple of \(\displaystyle \pi/2\). Then \(\displaystyle x=sin(t)\) is either 1 or -1.

 

[Deleted member 4993]

Find the area between the parametric curve, x(t)=cos(t), y(t)=sin^2(t) and the x-axis

<link removed>

My work shown in the link provided above without the bounds founded. Sorry for not rotating the image and poor quality. I need help trying to understand how to find the bounds for this equation.

Plot a x-y graph first.

That will give ideas to calculate the limits.

As you can see the x-y graph would be a parabola (y = 1 - x²)