Question

The integrand of the definite integral is a difference of two functions. Sketch the graph of each function and shade the region whose area is represented by the integral. $$ \int_{0}^{4}\left[(x+1)-\frac{1}{2} x\right] d x $$

Short answer

The area under the curve represented by the integral \(\int_{0}^{4}(x+1-\frac{1}{2}x) dx\) from 0 to 4 is 8 units square.

Step-by-step solution

Identify the Functions

The two functions in the integrand are \(f(x) = x+1\) and \(g(x) = -\frac{1}{2}x\). These will be graphed separately to later find the area between them from 0 to 4.

Sketch the Graphs of the Functions

Graph the first function \(f(x) = x + 1\) which is a line with a slope of 1 and crosses the y-axis at 1. Then, graph the second function \(g(x) = -\frac{1}{2}x\), a line with negative slope -1/2 and crosses the y-axis at 0. Both lines cross at the point (2,2).

Shade the Region

The area represented by the integral is the region between these two graphs from x=0 to x=4. Shade this region to show the area.

Evaluate the Definite Integral

The definite integral \(\int_{0}^{4}(x+1-\frac{1}{2}x) dx\) can now be solved. Simplify the integrand to \(\int_{0}^{4}(0.5x+1)dx\), separate and integrate term-by-term to obtain \(0.5*\frac{1}{2}*x^2+x |_{0}^{4}\) = 0.5*8+4-0 = 8.

Interpretation

The value obtained from the integral, which is 8, corresponds to the area of the shaded region on the graph. This area is enclosed by the two curves from x=0 to x=4 and is equal to the given integral value.