Final answer:
To calculate the area bound by x = 5, x = -5y, and y = 1 about y = 1, we sum the areas of two right triangles formed by the given lines and consider the effect of rotational axis on the final area calculation.
Step-by-step explanation:
To find the area of the region bounded by the lines x = 5, x = -5y, and y = 1 about the line y = 1, we can use the concept of definite integrals. This problem seems to be a piecewise area determination problem, requiring us to calculate the area of triangles or trapezoids formed by the intersecting lines and the axis of rotation.
The area under the line x = -5y between y = 0 and y = 1 is a right-angled triangle. The base of the triangle runs along the y-axis from 0 to 1, and the height along the x-axis from 0 to -5. Using the formula for the area of a triangle, we get 1/2 * base * height = 1/2 * 1 * 5 = 2.5.
Similarly, for the line x = 5, we get another triangle with the same dimensions but mirrored along the y-axis, also yielding an area of 2.5. Summing both areas gives us 2.5 + 2.5 = 5 square units. However, since we're rotating about y = 1, we should consider any adjustments needed for this rotation, which may involve the use of the washer or shell method if feasible.