Unit 11: Applications of Definite Integrals

Graphically, a definite integral represents the area under the curve of [latex]f(x)[/latex] between [latex]x=a[/latex] and [latex]x=b[/latex]. When [latex]f(x)[/latex] is above the [latex]x[/latex]-axis, this area is positive; when [latex]f(x)[/latex] is below the [latex]x[/latex]-axis, it is negative. This interpretation allows definite integrals to provide insight into physical quantities, like distance, area, and total change.

Practice Questions:

Evaluate the following indefinite integrals before referring to the video for comprehensive solutions.

Application 1: Area

Q1: Find the area under the curve [latex]y=x^2+1[/latex] over the interval [latex][-1,1][/latex].

Application 2: Position, Velocity, Acceleration

Q2: A motorboat is traveling at a constant velocity of 5.0 m/s when it starts to decelerate to arrive at the dock. Its acceleration is [latex]a(t)=-12-\frac{1}{4}t \; m/s^2[/latex]. (a) Find the velocity function of the motorboat. (b) At what time does the motorboat stop? (c) Find its position function. (d) What is the displacement of the motorboat from the time it begins to decelerate to when it stops?

Q3: A particle moves in a straight line with velocity [latex]v(t)=\sqrt{2t-2}[/latex] meters per second. At [latex]t=2[/latex]seconds, the particle’s distance from the starting point was [latex]6[/latex] meters in the positive direction. What is the particle’s position at [latex]t=5[/latex]seconds?