Question

Using integration, find the area of the region bounded by the line \[ y = 5x + 2, \] the \( x \)-axis, and the ordinates \( x = -2 \) and \( x = 2 \).

Hint:

To find the area between a curve and the \( x \)-axis, use \( A = \int_a^b f(x) dx \) and ensure proper limits.

Answer 1

Step 1: Set up the area integral. The given line equation is: \[ y = 5x + 2. \] The area enclosed between this line, the \( x \)-axis, and the vertical lines \( x = -2 \) and \( x = 2 \) is given by: \[ A = \int_{-2}^{2} (5x + 2) \,dx. \] Step 2: Evaluate the integral. \[ A = \int_{-2}^{2} (5x + 2) \,dx. \] \[ = \left[ \frac{5x^2}{2} + 2x \right]_{-2}^{2}. \] Step 3: Compute the definite integral. \[ = \left( \frac{5(2)^2}{2} + 2(2) \right) - \left( \frac{5(-2)^2}{2} + 2(-2) \right). \] \[ = \left( \frac{5(4)}{2} + 4 \right) - \left( \frac{5(4)}{2} - 4 \right). \] \[ = \left( 10 + 4 \right) - \left( 10 - 4 \right). \] \[ = 14 - 6 = 8. \] Final Answer: \[ A = 8 \text{ square units}. \]