Question

question what integral represents the area of the shaded region below? select the correct answer below: ∫₂⁶(7 - 1/x)dx ∫₄⁶(7 - 1/x)dx ∫₂⁶(7 - 2x)dx ∫₄⁶(7 - 2x)dx

Explanation:

Step1: Identify the lower and upper x - limits

The left - most x - value of the shaded region is \(x = 2\) and the right - most x - value is \(x = 6\).

Step2: Identify the function for the upper and lower bounds

The upper - bound of the shaded region is a horizontal line \(y = 2\) and the lower - bound is a line. We need to find the area between the curves. The area \(A\) between two curves \(y = f(x)\) and \(y = g(x)\) from \(x=a\) to \(x = b\) is given by \(A=\int_{a}^{b}(f(x)-g(x))dx\). Here, \(f(x)\) (upper curve) and \(g(x)\) (lower curve). If we assume the lower - bound line equation is \(y = 7 - 2x\) (by observing the slope and intercept if it is a linear function). The area of the shaded region is \(\int_{2}^{6}(2-(7 - 2x))dx=\int_{2}^{6}(2 - 7+2x)dx=\int_{2}^{6}(2x - 5)dx\). But if we consider the correct form based on the options and the general setup of finding area between a horizontal line and a non - horizontal line, the area of the region between \(y = 2\) and \(y=7 - 2x\) from \(x = 2\) to \(x = 6\) is \(\int_{2}^{6}(2-(7 - 2x))dx=\int_{2}^{6}(2x - 5)dx\). In the form of the given options, we note that the area between the horizontal line \(y = 2\) and the line \(y=7 - 2x\) from \(x = 2\) to \(x = 6\) is \(\int_{2}^{6}(2-(7 - 2x))dx\).

Answer:

\(\int_{2}^{6}(2-(7 - 2x))dx\)