- 15.1.1: (a) Estimate the volume of the solid that lies below the surface an...
- 15.1.2: If , use a Riemann sum with , to estimate the value of . Take the s...
- 15.1.3: (a) Use a Riemann sum with to estimate the value of , where . Take ...
- 15.1.4: (a) Estimate the volume of the solid that lies below the surface an...
- 15.1.5: A table of values is given for a function defined on (a) Estimate u...
- 15.1.6: A 20-ft-by-30-ft swimming pool is filled with water. The depth is m...
- 15.1.7: Let be the volume of the solid that lies under the graph of and abo...
- 15.1.8: The figure shows level curves of a function in the square . Use the...
- 15.1.9: A contour map is shown for a function on the square . (a) Use the M...
- 15.1.10: The contour map shows the temperature, in degrees Fahrenheit, at 4:...
- 15.1.11: Evaluate the double integral by first identifying it as the volume ...
- 15.1.12: Evaluate the double integral by first identifying it as the volume ...
- 15.1.13: Evaluate the double integral by first identifying it as the volume ...
- 15.1.14: The integral , where , represents the volume of a solid. Sketch the...
- 15.1.15: Use a programmable calculator or computer (or the sum command on a ...
- 15.1.16: Repeat Exercise 15 for the integral .
- 15.1.17: If is a constant function, , and , show that
- 15.1.18: Use the result of Exercise 17 to show that where .
Solutions for Chapter 15.1: DOUBLE INTEGRALS OVER RECTANGLES
Full solutions for Multivariable Calculus, | 7th Edition
Arc length formula
The length of an arc in a circle of radius r intercepted by a central angle of u radians is s = r u.
Component form of a vector
If a vector’s representative in standard position has a terminal point (a,b) (or (a, b, c)) , then (a,b) (or (a, b, c)) is the component form of the vector, and a and b are the horizontal and vertical components of the vector (or a, b, and c are the x-, y-, and z-components of the vector, respectively)
Continuous at x = a
lim x:a x a ƒ(x) = ƒ(a)
equation of a hyperbola
(x - h)2 a2 - (y - k)2 b2 = 1 or (y - k)2 a2 - (x - h)2 b2 = 1
equation of an ellipse
(x - h2) a2 + (y - k)2 b2 = 1 or (y - k)2 a2 + (x - h)2 b2 = 1
The left side of u(v + w) = uv + uw.
Focal length of a parabola
The directed distance from the vertex to the focus.
Future value of an annuity
The net amount of money returned from an annuity.
A shift of a graph to the left or right.
Connected subset of the real number line with at least two points, p. 4.
Inverse composition rule
The composition of a one-toone function with its inverse results in the identity function.
Linear regression equation
Equation of a linear regression line
The line segment through the foci of an ellipse with endpoints on the ellipse
Point-slope form (of a line)
y - y1 = m1x - x 12.
Positive linear correlation
See Linear correlation.
Principle of mathematical induction
A principle related to mathematical induction.
Product rule of logarithms
ogb 1RS2 = logb R + logb S, R > 0, S > 0,
A graph of a polar equation or r = a cos nu.
Stretch of factor c
A transformation of a graph obtained by multiplying all the x-coordinates (horizontal stretch) by the constant 1/c, or all of the y-coordinates (vertical stretch) of the points by a constant c, c, > 1.
Vector of length 1.