formula for area bounded by the curve x^2=4ay and x=mx+c

To find the area bounded by the parabola \( x^2 = 4ay \) and the straight line \( x = mx + c \), we need to follow a process that involves calculating the points of intersection and then setting up an appropriate integral.

### Steps:

1. **Find the Points of Intersection:**
   To find the points where the parabola \( x^2 = 4ay \) intersects the line \( x = mx + c \), substitute the equation of the line into the parabola.

   From \( x = mx + c \), solve for \( y \) in terms of \( x \):
   \[
   y = \frac{x - c}{m}
   \]

   Substituting this expression for \( y \) into the parabola's equation \( x^2 = 4ay \):
   \[
   x^2 = 4a\left( \frac{x - c}{m} \right)
   \]
   Simplifying, you get:
   \[
   x^2 = \frac{4a(x - c)}{m}
   \]
   Rearranging, we have a quadratic equation:
   \[
   mx^2 - 4ax + 4ac = 0
   \]
   Solve this quadratic equation to find the values of \( x \), which will give you the points of intersection.

2. **Set up the Definite Integral:**
   Once the limits of integration (points of intersection) are determined, the area between the two curves can be calculated by integrating the difference between the parabola and the line.

   The general formula for the area \( A \) between two curves \( y_1(x) \) and \( y_2(x) \) over the interval \( [x_1, x_2] \) is:
   \[
   A = \int_{x_1}^{x_2} \left[ y_1(x) - y_2(x) \right] dx
   \]
   Here, \( y_1(x) \) is the parabola equation \( y = \frac{x^2}{4a} \) and \( y_2(x) \) is the line equation \( y = \frac{x - c}{m} \).

   So the area \( A \) becomes:
   \[
   A = \int_{x_1}^{x_2} \left( \frac{x^2}{4a} - \frac{x - c}{m} \right) dx
   \]
   where \( x_1 \) and \( x_2 \) are the points of intersection obtained earlier.

3. **Evaluate the Integral:**
   After setting up the integral, solve it to find the exact area.

Would you like me to solve a specific example of this formula or explain a particular part further?

### 5 Related Questions:
1. How do you solve quadratic equations to find the points of intersection?
2. How do we determine whether the line intersects the parabola at one or two points?
3. How can we use definite integrals to find areas between curves?
4. What happens if the line does not intersect the parabola?
5. How do you interpret the geometry of this area on a graph?

### Tip:
Always sketch the graph of the curves to visually confirm the points of intersection and the area you are calculating.

Learn how to calculate the area bounded by the parabola x^2 = 4ay and the straight line x = mx + c. This involves solving quadratic equations, finding points of intersection, and using definite integrals. Ideal for students in Grades 11-12 or early college.

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