Properties of integrals Suppose that ∫ 1 4 f ( x ) d x = 6 , and ∫ 1 4 g ( x ) d x = 4 , and ∫ 3 4 f ( x ) d x = 2 . Evaluate the following integrals or state that there is not enough information. 43. ∫ 1 3 f ( x ) g ( x ) d x
Properties of integrals Suppose that ∫ 1 4 f ( x ) d x = 6 , and ∫ 1 4 g ( x ) d x = 4 , and ∫ 3 4 f ( x ) d x = 2 . Evaluate the following integrals or state that there is not enough information. 43. ∫ 1 3 f ( x ) g ( x ) d x
Solution Summary: The author explains that there is not enough information to evaluate displaystyle
Properties of integralsSuppose that
∫
1
4
f
(
x
)
d
x
=
6
, and
∫
1
4
g
(
x
)
d
x
=
4
, and
∫
3
4
f
(
x
)
d
x
=
2
. Evaluate the following integrals or state that there is not enough information.
43.
∫
1
3
f
(
x
)
g
(
x
)
d
x
With differentiation, one of the major concepts of calculus. Integration involves the calculation of an integral, which is useful to find many quantities such as areas, volumes, and displacement.
A ladder 27 feet long leans against a wall and the foot of the ladder is sliding away at a constant rate of 3 feet/sec. Meanwhile, a firefighter is climbing up the ladder at a rate of 2 feet/sec. When the firefighter has climbed up 6 feet of the ladder, the ladder makes an angle of л/3 with the ground. Answer the two related
rates questions below. (Hint: Use two carefully labeled similar right triangles.)
(a) If h is the height of the firefighter above the ground, at the instant the angle of the ladder with the ground is л/3, find dh/dt=
feet/sec.
(b) If w is the horizontal distance from the firefighter to the wall, at the instant the angle of the ladder with the ground is л/3, find dw/dt=
feet/sec.
Two cars start moving from the same point. One travels south at 60 mi/h and the other travels west at 25 mi/h. At what rate (in mi/h) is the distance between the cars increasing four hours later?
Step 1
Using the diagram of a right triangle given below, the relation between x, y, and z is
z²
= x²+
+12
x
Step 2
We must find dz/dt. Differentiating both sides and simplifying gives us the following.
2z
dz
dt
dx
2x.
+2y
dt
dx
dy
dz
x
+y
dt
dt
dt
2z
dy
dt
×
dx
(x+y
dt
dy
dt
An elastic rope is attached to the ground at the positions shown in the picture. The rope is being pulled up along the dotted line. Assume the units are meters.
9
ground level
Assume that x is increasing at a rate of 3 meters/sec.
(a) Write as a function of x: 0=
(b) When x=10, the angle is changing at a rate of
rad/sec.
(c) Let L be the the left hand piece of rope and R the right hand piece of rope. When x=10, is the rate of change of L larger than the rate of change of R?
○ Yes
○ No
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Numerical Integration Introduction l Trapezoidal Rule Simpson's 1/3 Rule l Simpson's 3/8 l GATE 2021; Author: GATE Lectures by Dishank;https://www.youtube.com/watch?v=zadUB3NwFtQ;License: Standard YouTube License, CC-BY