Integration

The equation of the curve is . Rearranging this equation, we get: We are asked to find the area bounded by the curve and the lines and . Step 1: Set up the integral for the area The area bounded by the curve and the lines is given by the integral of the difference between the x-values at the curve and the line , for between 0 and . The area is: Step 2: Perform the integration Now, integrate with respect to : Substitute the limits of integration: Thus, the area bounded by the curve and the lines is square units. Thus, the correct answer is .

We are tasked with evaluating the integral: Step 1: Use substitution to simplify the integral. Let us first consider the substitution , which will simplify the integral. To differentiate , we use the logarithmic differentiation technique. Differentiating both sides with respect to , we get: So, This allows us to rewrite the original integral: Step 2: Integrate. The integral is straightforward: Substitute back into the result: Thus, the integral simplifies to: Therefore, the correct answer is option (B)