Chapter 27

Find the area bounded by each curve, the \(x\) axis, and the given limits. $$y=3 x^{2}+2 x \quad \text { from } x=1 \text { to } 3$$

The current to a capacitor is given by \(i=2 t+3 .\) The initial charge on the capacitor is \(8.13 \mathrm{C} .\) Find the charge when \(t=1.00 \mathrm{s}.\)

A point in a machine has an initial displacement of \(12.6 \mathrm{cm}\) and has a velocity given by \(v=11.6 t+21.4 \mathrm{cm} / \mathrm{s} .\) (a) Write an equation for the displacement \(s\) and (b) evaluate it at \(t=7.00 \mathrm{s}\)

Find the area bounded by each curve, the \(x\) axis, and the given limits. $$y=4+2 x^{2}+2 x \quad \text { from } x=2 \text { to } 4$$

The current to a certain circuit is given by \(i=t^{2}+4 .\) If the initial charge is zero, find the charge at \(2.50 \mathrm{s}.\)

At a particular location in a mechanism, the initial displacement is 6.48 in. and the velocity is given by \(v=1.83+2.28 t^{2}\) in./s. (a) Write an equation for the displacement \(s\) and (b) evaluate it at \(t=4.00 \mathrm{s}\)

Find the area bounded by each curve, the \(x\) axis, and the given limits. $$y=3 \sqrt{x} \quad \text { from } x=1 \text { to } 5$$

Find the volume generated by rotating the first-quadrant area bounded by each set bout the Uses the disk or the shell method. $$y=\frac{x^{3 / 2}}{2} \text { and } x=2$$

The current to a certain capacitor is \(i=3.25+t^{3} .\) If the initial charge on the capacitor is \(16.8 \mathrm{C},\) find the charge when \(t=3.75 \mathrm{s}.\)

A car starts from rest and continues at a rate of \(v=\frac{1}{8} t^{2} \mathrm{ft} / \mathrm{s} .\) Find the function that relates the distance \(s\) the car has traveled to the time \(t\) in seconds. How far will the car go in 4 s?

Find the area bounded by each curve, the \(x\) axis, and the given limits. $$y=x+\sqrt{x} \quad \text { from } x=1 \text { to } 2$$

A body is moving at the rate \(v=\frac{3}{2} t^{2} \mathrm{m} / \mathrm{s} .\) Find the distance that it will move in \(t\) seconds if \(s=0\) when \(t=0\)

A 15.2 -F capacitor has an initial voltage of \(2.00 \mathrm{V}\). It is charged with a current given by \(i=t \sqrt{5+t^{2}} .\) Find the voltage across the capacitor at \(1.75 \mathrm{s}.\)

A pin on a robot arm has an initial velocity of \(2.58 \mathrm{ft} / \mathrm{s}\) and has an acceleration given by \(a=1.41 t^{2}+5.28 \mathrm{ft} / \mathrm{s}^{2} .\) (a) Write an equation for the velocity \(v\) and (b) evaluate it at \(t=1.00 \mathrm{s}\)

Find the first-quadrant area bounded by each curve and both coordinate axes. $$y=x^{3}-8 x^{2}+15 x$$

A point in a mechanism has an initial velocity of 44.3 in./s and has an acceleration given by \(a=52.6 t^{2}-41.1 t\) in. \(/ \mathrm{s}^{2} .\) (a) Write an equation for the velocity \(v\) and (b) evaluate it at \(t=2.00 \mathrm{s}\)

The voltage across a \(1.05-\mathrm{H}\) inductor is \(v=\sqrt{23 t} \mathrm{V} .\) Find the current in the inductor at 1.25 s if the initial current is zero.

Find the volume generated by rotating the first-quadrant area bounded by each set bout the Uses the disk or the shell method. $$x^{2 / 3}+y^{2 / 3}=1 \text { from } x=0 \text { to } 1$$

Find the first-quadrant area bounded by each curve and both coordinate axes. $$\sqrt{x}+\sqrt{y}=1$$

The voltage across a \(52.0-\mathrm{H}\) inductor is \(v=t^{2}-3 t\) V. If the initial current is \(2.00 \mathrm{A},\) find the current in the inductor at \(1.00 \mathrm{s}.\)

The acceleration of a point is given by \(a=4.00-t^{2} \mathrm{m} / \mathrm{s}^{2} .\) Write an equation for the velocity if \(v=2.00 \mathrm{m} / \mathrm{s}\) when \(t=3.00 \mathrm{s}\)

Find the area bounded by the curve \(10 y=x^{2}-80,\) the \(x\) axis, and the lines \(x=1\) and \(x=6\)

The voltage across a \(15.0-\mathrm{H}\) inductor is given by \(v=28.5+\sqrt{6 t} \mathrm{V} .\) Find the current in the inductor at \(2.50 \mathrm{s}\) if the initial current is \(15.0 \mathrm{A}.\)

Find the volume generated by rotating about the \(y\) axis the first-quadrant area bounded by each set of curves. \(y=x^{3},\) the \(y\) axis, and \(y=8\)

Find the area bounded by the curve \(y=x^{3},\) the \(x\) axis, and the lines \(x=-3\) and \(x=0\)

Find the volume generated by rotating about the \(y\) axis the first-quadrant area bounded by each set of curves. $$2 y^{2}=x^{3}, x=0, \text { and } y=2$$

Find only the portion of the area below the \(x\) axis. $$y=x^{3}-4 x^{2}+3 x$$

Find the volume generated by rotating about the \(y\) axis the first-quadrant area bounded by each set of curves. $$9 x^{2}+16 y^{2}=144$$

The acceleration of a falling body is \(a=-32.2 \mathrm{ft} / \mathrm{s}^{2} .\) Find the relation between \(s\) and \(t\) if \(s=0\) and \(v=20 \mathrm{ft} / \mathrm{s}\) when \(t=0\)

Find only the portion of the area below the \(x\) axis. $$y=x^{2}-4 x+3$$

A point starts from rest at the origin and moves along a curved path with \(x\) and \(y\) accelerations of \(a_{x}=2.00 \mathrm{cm} / \mathrm{s}^{2}\) and \(a_{y}=8.00 t \mathrm{cm} / \mathrm{s}^{2} .\) Write expressions for the \(x\) and \(y\) components of velocity.

Find the first-quadrant area bounded by the given curve, the \(y\) axis, and the given lines. \(y=x^{2}+2\) from \(y=3\) to 5

Find the volume generated by rotating about the \(y\) axis the first-quadrant area bounded by each set of curves. $$y^{2}=4 x \text { and } y=4$$

A point starts from rest at the origin and moves along a curved path with \(x\) and \(y\) accelerations of \(a_{x}=5.00 t^{2} \mathrm{cm} / \mathrm{s}^{2}\) and \(a_{y}=2.00 t \mathrm{cm} / \mathrm{s}^{2} .\) Find the \(x\) and \(y\) components of velocity at \(t=10.0 \mathrm{s}\)

Find the first-quadrant area bounded by the given curve, the \(y\) axis, and the given lines. \(8 y^{2}=x\) from \(y=0\) to 10

Find the volume generated by rotating about the indicated axis the first- quadrant area bounded by the given pair of curves. \(y=3 x^{2}\) and \(x=2,\) about the \(y\) axis.

Find the first-quadrant area bounded by the given curve, the \(y\) axis, and the given lines. \(y^{3}=4 x\) from \(y=0\) to 4

Find the volume generated by rotating about the indicated axis the first- quadrant area bounded by the given pair of curves. \(y=2 \sqrt{x}\) and \(x=3,\) about the \(y\) axis.

A point starts from (9,1) with initial velocities of \(v_{x}=6.00 \mathrm{cm} / \mathrm{s}\) and \(v_{y}=2.00 \mathrm{cm} / \mathrm{s}\) and moves along a curved path. It has \(x\) and \(y\) accelerations of \(a_{x}=3 t\) and \(a_{y}=2 t^{2} .\) Find the \(x\) and \(y\) components of velocity at \(t=15.0 \mathrm{s}\)

Find the volume generated by rotating about the indicated axis the first- quadrant area bounded by the given pair of curves. \(y=2 x^{3},\) the \(y\) axis, and \(y=7,\) about the \(x\) axis.

A wheel starts from rest and accelerates at \(3.00 \mathrm{rad} / \mathrm{s}^{2} .\) Find the angular velocity after \(12.0 \mathrm{s}\)

Find the volume generated by rotating about the indicated axis the first- quadrant area bounded by the given pair of curves. \(y=3 \sqrt{x},\) and \(y=2,\) about the \(x\) axis.

Find the area bounded by the curve \(y^{2}=x^{3}\) and the line \(x=4\)

Find the volume generated by rotating about the indicated axis the first- quadrant area bounded by each set of curves. $$x=4 \text { and } y^{2}=x^{3}, \text { about } x=4$$

A link in a mechanism rotating with an angular velocity of \(3.00 \mathrm{rad} / \mathrm{s}\) is given an acceleration of \(5.00 \mathrm{rad} / \mathrm{s}^{2}\) at \(t=0 .\) Find the angular velocity after \(20.0 \mathrm{s}\)

A pulley in a magnetic tape drive is rotating at \(1.25 \mathrm{rad} / \mathrm{s}\) when it is given an acceleration of \(7.24 \mathrm{rad} / \mathrm{s}^{2}\) at \(t=0 .\) Find the angular velocity at \(2.00 \mathrm{s}\)

Find the area bounded by the parabola \(y=3-x^{2}\) and the line \(y=x+1\)

Find the volume generated by rotating about the indicated axis the first- quadrant area bounded by each set of curves. $$y^{2}=x^{3} \text { and } y=8, \text { about } y=9$$

Find the area bounded by the curves \(y^{2}=4 x\) and \(2 x-y=4\)

Find the volume generated by rotating about the indicated axis the first- quadrant area bounded by each set of curves. $$y=4 \text { and } y=4+6 x-2 x^{2}, \text { about } y=4$$