Question

Find the area of the region under the graph of the function f on the interval [-1, 4] y=2x+3

Answer 1

The required area is shown in darkest shading above the x-axis.

There are two ways to calculate the area: calculus and geometry.

CALCULUS

Area=∫[-1,4](2x+3)dx=(x²+3x)[-1,4]=(16+12)-(1-3)=28+2=30 sq units.

GEOMETRY

Difference in area of two right triangles: 

Large triangle has height 11 and base=(4-(-1.5))=4+1.5=5.5. Area=½(5.5)(11)=30.25 sq units.

Small triangle has height 1 and base=0.5. Area=½(0.5)(1)=0.25 sq units.

Difference=30.25-0.25=30 sq units.