Calculating Volumes of Bounded Regions (Curve, Origin and Interval)

1. Sketch the region bounded between the given curves and then find the area of each region
a) {see attachment}
b) Find the area of the region than contains the origin and is bounded by the line {see attachment}

2. Sketch the given region and then find the volume of the solid whose base is the given region and which has the property that each cross section {see attachment} is a square.
a) The region bounded by the x-axis and the semi-circle {see attachment}
b) The region bounded by {see attachment} and below by the x-axis on the interval {see attachment}

1. Solve the following differential equation:
-2yy' +3x^2 SQRT(4-y^2) =5x^2 SQRT(4-y^2) , -2 < y < +2
2. Let f(x) = ax^2 , a>0 , and g(x) = x^3
Find the value of a which yields an area of PI (i.e. 3.14159) for region bounded by figure, y-axis and line x=1.

Calculate the area under the curve y=1/(x^2) above the x-axis on the interval [1, positive infinity].
keywords: finding, find, calculate, calculating, determine, determining, verify, verifying

Consider the titration of 50.0 mL of 0.0110 M Y^3+ (y=yttrium) with 0.0220 M EDTA at pH 5.00. Calculate pY^3+ at the following volumes of added EDTA and sketch the titration curve: (a) 0mL (b) 10.0 mL (c) 25 mL (d) 30.0 mL.

1. Solve the following differential equation:
-2yy' +3x² √(4-y²) =5x² √(4-y²) , -2< y<+2
2. A calculus instructor has determined that the arc of an individual diving into a swimming pool is defined by the function, f(x) = sin (.4x). Determine how far the diver has traversed in his dive as he passes thr

Let f(x), g(x) be functions defined on a closed bounded interval [a, b] such that the following conditions hold:
g is differentiable on [a, b].
There are positive constants a, b such that g(x) = a*f(x) - b*(dg/dx).
f(x) > 0 for all x in [a, b]
g(x) >= 0 for all x in [a, b]
g(a) > 0
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Please help with the following problem. Provide step by step calculations for each.
The average value of f(x) = 1/x on the interval [4, 16] is
(ln 2)/3
(ln 2)/6
(ln 2)/12
3/2
0
1
none of these
Find the area, in square units, of the region b

1. A function f:R-->R is said to be periodic if there is a number p > 0 such that f(x) = f(x+p) for all xER . Show that a continuous periodic function on R is boundedand uniformly continuous.