§6.1 Using definite integrals to find area and length

Introduction

The Area Between Two Curves

Finding Area with Horizontal Slices

Finding the length of a curve

Consider the functions given by $f(x) = 5-(x-1)^2$ and $g(x) = 4-x$.

Use algebra to find the points where the graphs of $f$ and $g$ intersect.

Find the area bounded between the graphs of $f(x) = (x-1)^2 + 1$ and $g(x) = x+2$.

The finite region bounded by $y = \sqrt{x}$ and $y = \frac{1}{4}x$.

The finite region bounded by $y = 12-2x^2$ and $y = x^2 - 8$.

The area bounded by the $y$-axis, $f(x) = \cos(x)$, and $g(x) = \sin(x)$, where we consider the region formed by the first positive value of $x$ for which $f$ and $g$ intersect.

The finite regions between the curves $y = x^3-x$ and $y = x^2$.

Find the area of the region bounded by the parabola $x = y^2 - 1$ and the line $y = x-1$, shown at left in Figure 6.1.

The finite region bounded by $x=y^2$ and $x=6-2y^2$.

The finite region bounded by $x=1-y^2$ and $x = 2-2y^2$.

The area bounded by the $x$-axis, $y=x^2$, and $y=2-x$.

The finite regions between the curves $x=y^2-2y$ and $y=x$.

Use the definition and appropriate computational technology to determine the arc length along $y = x^2$ from $x = -1$ to $x = 1$.

Find the arc length of $y = \sqrt{4-x^2}$ on the interval $-2 \le x \le 2$. Find this value in two different ways: (a) by using a definite integral, and (b) by using a familiar property of the curve.

Determine the arc length of $y = xe^{3x}$ on the interval $[0,1]$.

Will the integrals that arise calculating arc length typically be ones that we can evaluate exactly using the First FTC, or ones that we need to approximate? Why?

A moving particle is traveling along the curve given by $y = f(x) = 0.1x^2 + 1$, and does so at a constant rate of 7 cm/sec, where both $x$ and $y$ are measured in cm (that is, the curve $y = f(x)$ is the path along which the object actually travels; the curve is not a “position function”). Find the position of the particle when $t = 4$ sec, assuming that when $t = 0$, the particle's location is $(0,f(0))$.