Find the Area Between the Curves x=3t , y=3-t , t=5

Problem

x=3*t,y=3−t,t=5

Solution

1. Identify the task as finding the area between a parametric curve and the axes or a specific boundary. Given the single value t=5 and the equations for x and y we find the area under the curve from the starting point (where y=0 to the point where t=5

2. Determine the limits of integration for t The curve intersects the xaxis when y=0

3−t=0

t=3

The interval for t is [3,5]

3. Set up the integral for the area A using the parametric formula A=(∫_(t_1)^(t_2))(y(t)d(x)/d(t)*d(t))

4. Calculate the derivative of x with respect to t

d(x)/d(t)=3

5. Substitute the expressions into the integral.

A=(∫_3^5)((3−t)*(3)*d(t))

A=(∫_3^5)((9−3*t)*d(t))

6. Evaluate the definite integral.

A=[9*t−(3*t2)/2]53

A=(9*(5)−(3*(5)2)/2)−(9*(3)−(3*(3)2)/2)

A=(45−75/2)−(27−27/2)

A=15/2−27/2

A=−6

7. Take the absolute value since area must be positive.

|A|=6

Final Answer

Area=6

Want more problems? Check here!