Integrating piecewise continuous functions Recall that the floor function ⌊ x ⌋ is the greatest integer less than or equal to x and that the ceiling function ⌈ x ⌉ is the least integer greater than or equal to x . Use the result of Exercise 88 and the graphs to evaluate the following integrals. 91. ∫ 1 5 x ⌊ x ⌋ d x
Integrating piecewise continuous functions Recall that the floor function ⌊ x ⌋ is the greatest integer less than or equal to x and that the ceiling function ⌈ x ⌉ is the least integer greater than or equal to x . Use the result of Exercise 88 and the graphs to evaluate the following integrals. 91. ∫ 1 5 x ⌊ x ⌋ d x
Solution Summary: The author evaluates the value of displaystyle 'underset' by using the result of exercise 88 and the given graph.
Integrating piecewise continuous functions Recall that the floor function
⌊
x
⌋
is the greatest integer less than or equal to x and that the ceiling function
⌈
x
⌉
is the least integer greater than or equal to x. Use the result of Exercise 88 and the graphs to evaluate the following integrals.
91.
∫
1
5
x
⌊
x
⌋
d
x
Definition Definition Group of one or more functions defined at different and non-overlapping domains. The rule of a piecewise function is different for different pieces or portions of the domain.
Electric charge is distributed over the triangular region D shown below so that the charge density at (x, y)
is σ(x, y) = 4xy, measured in coulumbs per square meter (C/m²). Find the total charge on D. Round
your answer to four decimal places.
1
U
5
4
3
2
1
1
2
5
7
coulumbs
Let E be the region bounded cone z = √√/6 - (x² + y²) and the sphere z = x² + y² + z² . Provide an
answer accurate to at least 4 significant digits. Find the volume of E.
Triple Integral
Spherical Coordinates
Cutout of sphere is for visual purposes
0.8-
0.6
z
04
0.2-
0-
-0.4
-0.2
04
0
0.2
0.2
x
-0.2
04 -0.4
Note: The graph is an example. The scale and equation parameters may not be the same for your
particular problem. Round your answer to 4 decimal places.
Hint: Solve the cone equation for phi.
* Oops - try again.
The temperature at a point (x,y,z) of a solid E bounded by the coordinate planes and the plane
9.x+y+z = 1 is T(x, y, z) = (xy + 8z +20) degrees Celcius. Find the average temperature over
the solid. (Answer to 4 decimal places).
Average Value of a function
using 3 variables
z
1-
y
Hint: y = -a·x+1
* Oops - try again.
x
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