Math 165 – worksheet for ch. 5, Integration – solutions
... Solution Cut the region into two triangles from x = 0 to x = 1.5 and x = 1.5 to x = 3, then a rectangle from x = 3 to x = 10 with a semicircle on top that has diameter 4, and another triangle from x = 10 to x = 13. The first two triangles have area 9/4 each but they cancel each other out (for the in ...
... Solution Cut the region into two triangles from x = 0 to x = 1.5 and x = 1.5 to x = 3, then a rectangle from x = 3 to x = 10 with a semicircle on top that has diameter 4, and another triangle from x = 10 to x = 13. The first two triangles have area 9/4 each but they cancel each other out (for the in ...
Lecture Notes for Section 6.1
... Section 6.1: Review of Integration Formulas and Techniques Big idea: With some creative algebra, you can do a lot of “new-looking” integrals by manipulating the integrand to match integral formulas from Calculus 1. Big skill: You should be able to manipulate the integrands of the integrals in this s ...
... Section 6.1: Review of Integration Formulas and Techniques Big idea: With some creative algebra, you can do a lot of “new-looking” integrals by manipulating the integrand to match integral formulas from Calculus 1. Big skill: You should be able to manipulate the integrands of the integrals in this s ...
MATH 1920 --- CALCULUS II COURSE SYLLABUS INSTRUCTOR
... 4. Express the length of a curve as a (Riemann) sum of linear segments, convert to definite integral form and compute its value. 5. Express the surface area of revolution of a function’s graph around a given axis as a (Riemann) sum of rings, convert to definite integral form and compute its value. 6 ...
... 4. Express the length of a curve as a (Riemann) sum of linear segments, convert to definite integral form and compute its value. 5. Express the surface area of revolution of a function’s graph around a given axis as a (Riemann) sum of rings, convert to definite integral form and compute its value. 6 ...
CalculusLecture-384H.pdf
... Similarly, there are higher analogues of integrals and dierentials. We won't go into detail here, but in a nutshell, the double integral over a region R in the xy -plane of a continuous function f (x, y) de ned on that region is a number ZZ f (x, y) dA R ...
... Similarly, there are higher analogues of integrals and dierentials. We won't go into detail here, but in a nutshell, the double integral over a region R in the xy -plane of a continuous function f (x, y) de ned on that region is a number ZZ f (x, y) dA R ...
The Fundamental Theorem of Calculus.
... The two main concepts of calculus are integration and differentiation. The Fundamental Theorem of Calculus (FTC) says that these two concepts are essentially inverse to one another. The fundamental theorem states that if F has a continuous derivative on an interval [a, b], then Z b F 0 (t)dt = F (b) ...
... The two main concepts of calculus are integration and differentiation. The Fundamental Theorem of Calculus (FTC) says that these two concepts are essentially inverse to one another. The fundamental theorem states that if F has a continuous derivative on an interval [a, b], then Z b F 0 (t)dt = F (b) ...
MATHEMATICS 2030
... derivatives), directional derivatives, gradient, higher partial derivatives and Clairaut’s Theorem, optimization (maxima and minima) c. Vector fields, gradient (conservative) vector fields, conditions for conservativity over domains in R2 and R3, calculation of the primitives of a conservative vecto ...
... derivatives), directional derivatives, gradient, higher partial derivatives and Clairaut’s Theorem, optimization (maxima and minima) c. Vector fields, gradient (conservative) vector fields, conditions for conservativity over domains in R2 and R3, calculation of the primitives of a conservative vecto ...
Solutions - Penn Math
... You could also do this problem without using the fundamental theorem of line integrals, just by using the definition of a line integral. ...
... You could also do this problem without using the fundamental theorem of line integrals, just by using the definition of a line integral. ...
Math 125 – Section 02
... arithmetic) to get the answer, explain what you did and why you did it. Point values for each part are given in brackets. [22 total] NO WORK = NO CREDIT!! Answer the following TRUE/FALSE questions. Defend your answer, if it is true, explain why, if it is false, give an example showing that it is fal ...
... arithmetic) to get the answer, explain what you did and why you did it. Point values for each part are given in brackets. [22 total] NO WORK = NO CREDIT!! Answer the following TRUE/FALSE questions. Defend your answer, if it is true, explain why, if it is false, give an example showing that it is fal ...
Solutions to some problems (Lectures 15-20)
... (a) By finding the potential functions, show that each of the vector fields ~ H ~ is a gradient vector field. F~ , G, ~ H ~ around the unit circle in the xy-plane, (b) Find the line integrals of F~ , G, centered at the origin, and traversed counterclockwise. (c) For which of the three vector fields ...
... (a) By finding the potential functions, show that each of the vector fields ~ H ~ is a gradient vector field. F~ , G, ~ H ~ around the unit circle in the xy-plane, (b) Find the line integrals of F~ , G, centered at the origin, and traversed counterclockwise. (c) For which of the three vector fields ...
The Fundamental Theorem of Calculus and Integration
... Here, n is the number of rectangles used in the approximation, xi is the xvalue at the left-hand edge of each rectangle, and ∆x is the width of each rectangle. We concluded the lecture by saying that, in practice, we never compute this limit directly (though in some cases it is possible, and not too ...
... Here, n is the number of rectangles used in the approximation, xi is the xvalue at the left-hand edge of each rectangle, and ∆x is the width of each rectangle. We concluded the lecture by saying that, in practice, we never compute this limit directly (though in some cases it is possible, and not too ...
Fundamental theorem of calculus part 2
... Increasing/decreasing functions and their derivatives ...
... Increasing/decreasing functions and their derivatives ...
The Fundamental Theorem of Calculus
... Discovered independently by Gottfried Liebnitz and Isaac Newton Informally states that differentiation and definite integration are inverse operations. ...
... Discovered independently by Gottfried Liebnitz and Isaac Newton Informally states that differentiation and definite integration are inverse operations. ...