Question

Find the area bounded between two curves: \[ x^2 + y^2 = 25 \quad \text{and} \quad y = |x - 1|. \]

Hint:

When calculating the area between curves, always set up the integral by subtracting the lower curve from the upper curve. Pay attention to the points of intersection and break the integral into parts if needed.

Answer 1

The first equation \( x^2 + y^2 = 25 \) represents a circle with a radius of 5, centered at the origin. The second equation \( y = |x - 1| \) represents a V-shaped graph that has a vertex at \( (1, 0) \).

Step 1: Determine points of intersection To find the points of intersection, substitute \( y = |x - 1| \) into \( x^2 + y^2 = 25 \): - For \( x \geq 1 \), \( y = x - 1 \). - For \( x<1 \), \( y = 1 - x \). 

Case 1: \( x \geq 1 \) Substitute \( y = x - 1 \) into the circle equation: \[ x^2 + (x - 1)^2 = 25. \] Simplify: \[ x^2 + (x^2 - 2x + 1) = 25 \quad \Rightarrow \quad 2x^2 - 2x + 1 = 25 \quad \Rightarrow \quad 2x^2 - 2x - 24 = 0. \] 

Solve the quadratic equation: \[ x^2 - x - 12 = 0. \] The solutions are: \[ x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(1)(-12)}}{2(1)} = \frac{1 \pm \sqrt{49}}{2} = \frac{1 \pm 7}{2}. \] So, \( x = 4 \) or \( x = -3 \). 

Case 2: \( x<1 \) Substitute \( y = 1 - x \) into the circle equation: \[ x^2 + (1 - x)^2 = 25. \] Simplify and solve similarly. 

Step 2: Calculate the area The area between these two curves can be computed by integrating the difference between the top curve \( y = |x - 1| \) and the bottom curve \( y = \sqrt{25 - x^2} \). 

This is done by setting up the integral over the range of \( x \) where the curves intersect. The integral for the area is: \[ \text{Area} = \int_{-3}^{4} \left( \sqrt{25 - x^2} - |x - 1| \right) dx. \] 

This will give the total area between the two curves.