Question

The area in the first quadrant bounded by the function \(y = (8-x)\) and the coordinate axes is ______ square units (answer in integer).

Hint:

The line y = 8 - x meets the axes at (8,0) and (0,8), so the bounded region in the first quadrant is just a right triangle; use either integration or the (1/2) × base × height formula.

Answer 1

Correct Answer: 32

Step 1: Identify the boundary curve.
The function y = 8 - x is a straight line, not a curve with any bend, since it is degree one in x. Because the region is bounded by this line together with the coordinate axes in the first quadrant, the region is simply a triangle, and we can either integrate or use plain geometry.

Step 2: Find where the line meets each axis.
Putting x = 0 gives y = 8, so the line crosses the y-axis at (0, 8). Putting y = 0 gives x = 8, so the line crosses the x-axis at (8, 0). For x between 0 and 8, y = 8 - x stays non-negative, so the enclosed region lies entirely in the first quadrant.

Step 3: Recognize the shape of the bounded region.
The region bounded by the line and the two axes has vertices at the origin (0,0), the x-intercept (8,0) and the y-intercept (0,8). This is a right-angled triangle with the right angle at the origin, legs of length 8 along the x-axis and 8 along the y-axis.

Step 4: Compute the area, first by integration.
Area = ∫ from 0 to 8 of (8 - x) dx = [8x - x²/2] evaluated from 0 to 8 = (8 × 8 - 8²/2) - 0 = (64 - 32) = 32 square units.

Step 5: Cross-check using the triangle area formula.
Area of right triangle = (1/2) × base × height = (1/2) × 8 × 8 = 32 square units, which matches the integration result exactly.

Step 6: State the final answer.
The area of the region in the first quadrant bounded by y = 8 - x and the coordinate axes is 32 square units, an exact integer as expected for this simple linear boundary.