Use trapezoids to approximate the area under the curve y=f(x)=5sin(πx)+2.5cos(4πx) on the interval [0,1] using 10 equal subdivisions.

Question

Answers can be either: 3.157, 3.183, 6.314, 63.138, or 3.078

Answer 1

The table shows f(x) for values of x from 0 to 1, and the graph shows the plot of (x,f(x)) for values of x=0, 0.1, 0.2, etc.

Because of symmetry, it’s only necessary to find the areas of the trapezoids between 0 and 0.5. This is half the total area. Each trapezoid is made up of a triangle and a rectangle. So we can work out the areas as follows. Call the areas A to E for [0,0.1] to [0.4,0.5]:

A: 0.1*2.3176+0.05(2.5-2.3176)=0.23176+0.00912=0.24088

B: 0.1*0.9164+0.05(2.3176-0.9164)=0.09164+0.07006=0.16170

C: 0.1*0.9164+0.05(2.0225-0.9164)=0.09164+0.055305=0.146945

D: 0.1*2.0225+0.05(5.5278-2.0225)=0.20225+0.175265=0.377515

E: 0.1*5.5278+0.05(7.5-5.5278)=0.55278+0.09861=0.65139

Add these together: 1.57843. Total area: 3.157 approx, answer option 1.