VIEWS: 9 PAGES: 7 POSTED ON: 7/19/2011
Calc 3 Lecture Notes Section 14.2 Page 1 of 7 Section 14.2: Line Integrals Big idea: Line integrals are used to compute how a given quantity “adds up” along a curved path. Big skill: You should be able to compute line integrals (using the evaluation theorem). Introductory examples: x kg 1. Compute the mass of a baseball bat with linear density x 0.5 for 0 x m m 1 m. (Note that in this example, we are adding up a quantity over a curve, which is simply a straight segment of length 1 m.) 2. Compute the mass of a helix of radius 1 m and that has 5 turns along 1 m of its axis, and a linear mass density that increases linearly from 1 g/m to 26 g/m over its length. Notice that the total arc length of this helix is: 2 1m 0.2m 10 2 0.01m 31.432m , so we can write the linear mass 2 2 5 25s g density as a function of arc length s as: s 1 for s over the interval 31.432m m 0 s 31.432 m. z x y Note that in this example, we had to make an effort to relate the density to the arc length. This is not usually how things are done. Instead, we parameterize the curve and leave the integrand alone… Calc 3 Lecture Notes Section 14.2 Page 2 of 7 Definition 2.1: Line Integral with Respect to Arc Length The line integral of f(x, y, z) with respect to arc length along the oriented curve C in three- dimensional space, written as f x, y, z ds is defined by C n f x, y, z ds lim f x *, y *, z * s , C P 0 i 1 i i i i provided the limit exists and is the same for all choices of evaluation points. Theorem 2.1: Evaluation Theorem (for Line Integrals) If f(x, y, z) is continuous in a region D containing a curve C, C can be described parametrically by (x(t), y(t), z(t)) for a t b, and x(t), y(t), and z(t) have continuous first derivatives, then b f x, y, z ds f x t , y t , z t x t y t z t dt . 2 2 2 C a Likewise, if f(x, y) is continuous in a region D containing a curve C, C can be described parametrically by (x(t), y(t)) for a t b, and x(t) and y(t) have continuous first derivatives, then b f x, y ds f x t , y t x t y t dt . 2 2 C a Proof: Calc 3 Lecture Notes Section 14.2 Page 3 of 7 Practice: 25 z g 1. Compute the mass of the helix from page 1 given that x, y, z 1 . m m 2. Evaluate x y z ds , given a curve C specified by x = cos(t), y = sin(t), and z = cos(t) C for 0 t 2. 3. Evaluate xyds , given a curve C specified by the quarter ellipse in the first quadrant of C the x-y plane with a semi-major axis of length 2 aligned along the x-axis and semi-minor axis of 1. Calc 3 Lecture Notes Section 14.2 Page 4 of 7 A curve C is called smooth if it can be described parametrically by (x(t), y(t), z(t)) for a t b, x(t), y(t), and z(t) have continuous first derivatives, and x t y t z t 0 on the 2 2 2 interval [a, b]. For example, the curve C = (t2, t3, t2) is not smooth on the interval [-1, 1] because 2t 3t 2t 0 when t = 0: 2 2 2 z x y This means that the line integral 1 f x, y, z ds f x t , y t , z t x t y t z t dt is undefined, but if we 2 2 2 C 1 break up the curve into two subpieces C = C1 C2 over the intervals [-1,0] and [0, 1], then we can break up the integral into two subpieces as well. A curve that can be subdivided into smooth subpieces is called piecewise smooth. Theorem 2.2: Line Integrals on a Piecewise Smooth Curve If f(x, y, z) is continuous in a region D containing an oriented curve C, and C is piecewise- smooth with C = C1 C2 … Cn and all the Ci are smooth and the terminal point of Ci is the initial point of Ci+1 for all i = 1, 2, … n-1, THEN f x, y, z ds f x, y, z ds C C AND f x, y, z ds f x, y, z ds f x, y, z ds C C1 C2 f x, y, z ds Cn Practice: 4. Evaluate xyds , given a curve C specified by a sector of outer radius 2 and inner radius C 1 and angular extent of /2 symmetric to the x-axis. Calc 3 Lecture Notes Section 14.2 Page 5 of 7 Geometric interpretation of the line integral: b Just as f x dx is the area bounded by x = a, x = b, y = 0 and y = f(x), a b f x, y ds f x t , y t ds is the area of the “vertical” surface bounded by the “vertical” C a line through (x(a), y(a)), the vertical line through (x(b), y(b)), the parametric curve (x(t), y(t), 0), and the parametric curve ( x(t), y(t), f(x(t), y(t)) ). Theorem 2.3: Arc Length of a Curve from a Line Integral For any piecewise-smooth curve C, 1ds gives the arc length of the curve C. C Next, we worry about evaluating a line integral where the integrand is the dot product of a vector field and the differential tangent vector to the curve, as arises in the computation of work: Calc 3 Lecture Notes Section 14.2 Page 6 of 7 Definition 2.2: Component-Wise Line Integrals The line integral of f(x, y, z) with respect to x along the oriented curve C in three-dimensional space is written as f x, y, z dx and is defined by: C n f x, y, z dx lim f xi *, yi *, zi * xi , P 0 C i 1 provided the limit exists and is the same for all choices of evaluation points. The line integral of f(x, y, z) with respect to y along the oriented curve C in three-dimensional space is written as f x, y, z dy and is defined by: C n f x, y, z dy lim f xi *, yi *, zi * yi , P 0 C i 1 provided the limit exists and is the same for all choices of evaluation points. The line integral of f(x, y, z) with respect to z along the oriented curve C in three-dimensional space is written as f x, y, z dz and is defined by: C n f x, y, z dz lim f xi *, yi *, zi * zi , P 0 C i 1 provided the limit exists and is the same for all choices of evaluation points. Theorem 2.4: Evaluation Theorem (for Component-Wise Line Integrals) If f(x, y, z) is continuous in a region D containing a curve C, C can be described parametrically by (x(t), y(t), z(t)) for a t b, and x(t), y(t), and z(t) have continuous first derivatives, then b f x, y, z dx f x t , y t , z t x t dt C a b f x, y, z dy f x t , y t , z t y t dt . C a b f x, y, z dz f x t , y t , z t z t dt C a Calc 3 Lecture Notes Section 14.2 Page 7 of 7 Theorem 2.5: Evaluation Theorem (for Component-Wise Line Integrals) If f(x, y, z) is continuous in a region D containing an oriented curve C, then: If C is piecewise-smooth: f x, y, z dx f x, y, z dx C C f x, y, z dy f x, y, z dy C C f x, y, z dz f x, y, z dz C C If C is piecewise-smooth and forms a closed loop: f x, y, z dx f x, y, z dx f x, y, z dx C C1 C1 f x, y, z dx Cn f x, y, z dy f x, y, z dy f x, y, z dy C C1 C1 f x, y, z dy Cn f x, y, z dz f x, y, z dz f x, y, z dz C C1 C1 f x, y, z dz Cn Consequence: W F x, y, z dr C