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Definite integral of a constant function via rectangle area

Consider the constant function f(x)=5f(x) = 5 defined on the interval [1,6][1, 6].

The graph of f(x)=5f(x) = 5 on [1,6][1,6] forms a rectangle bounded above by the line y=5y = 5, below by the x-axis, and on the sides by x=1x=1 and x=6x=6.

Definite integral of a constant function via rectangle area graph

Using the fact that this region is a rectangle, evaluate ∫165 dx\displaystyle\int_1^6 5 \, dx by computing its area directly (no antiderivative needed).

Hint

The region under a constant function over an interval is a rectangle. Its area is base times height, where the base is the length of the interval [1,6][1,6] and the height is the constant value of ff.

Step-by-step solution

The region bounded by f(x)=5f(x) = 5, the x-axis, and the vertical lines x=1x=1 and x=6x=6 is a rectangle. Its height is the constant function value, 55, and its base is the length of the interval, 6−1=56 - 1 = 5.

The area of a rectangle is base times height, so the area is 5×5=255 \times 5 = 25. Since f(x)=5>0f(x) = 5 > 0 on all of [1,6][1,6], this region lies entirely above the x-axis, so the signed area equals the ordinary area. Therefore ∫165 dx=25\int_1^6 5\,dx = 25.

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