Consider this curve.

Recall that the area under a function curve on the interval from a to b is given by the definite integral of the function, with a and b as the limits of integration.

Now, suppose this curve is described by two parametric equations.

The area under the curve is determined as t increases from a to b.

The area under a curve is given by the definite integral in parametric form.

Consider the curve described by these parametric equations.

We wish to determine the area under the curve from t equals 1 to t equals 2.

Consider the parametric curve shown in this graph.

Use the slider to rotate the curve about the y-axis.

If we revolve the parametric curve about the y-axis, we get a surface of revolution that is a hemisphere.

We already know an equation for determining the surface area when y is a known function of x.

Using the chain rule for integrals, we can derive an equation for the surface area in parametric form.

To determine the surface area of the hemisphere, we substitute the parameters into the equation.

Which of the following integrals gives the area under the curve in terms of the parametric equations? Click the "Submit" button after selecting your answer.

What is the area under the curve? Click the "Submit" button after selecting your answer.

What type of surface of revolution is created? Click the "Submit" button after selecting your answer.

What is the surface area of the hemisphere? Click the "Submit" button after selecting your answer.

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