Exam 7: Area of a Region Between Two Curves

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the given lines. $x = y ^ { 2 } , x = 8 y - y ^ { 2 }$ (i) $y$ -axis; (ii) the line $x = 18$

Find M_x,M_y , and ( $\bar{x}$ , $\bar {y}$ ) for the lamina of uniform density $\rho$ text bounded by the graphs

Find the area of the region bounded by the graphs of the algebraic functions. f(x)=-20x g(x)=0

Find the area of the region bounded by the graphs of the algebraic functions. $f ( y ) = y ^ { 2 } + 12 , \quad g ( y ) = 0 , \quad y = - 12 , \quad y = 13$

Find the area of the region bounded by the graphs of the algebraic functions. f(x)= g(x)=x-8

Use the shell method to set up and evaluate the integral that gives the volume of the solid generated by revolving the plane region about the x-axis. $y = x , y = 0 , x = 20$

Find M_x for the lamina of uniform density $\rho$ bounded by the graphs of the equations $y = 17 x ^ { 2 }$ and $y = 17 x ^ { 3 }$ .

A rectangular plate of height h feet and base b feet is submerged vertically in a tank of fluid that weighs $w$ pounds per cubic foot. The center of the plate is $k$ feet below the surface of the fluid. The fluid force on the surface of the plate is given by $F = w k h b$ Find the fluid force on the rectangular plate as shown in the figure given $x = 6$ feet and $y = 4$ feet. Round your answer to one decimal place.

Find the fluid force on the vertical side of the tank, where the dimensions are given in feet. Assume that the tank is full of water. Note: The density of water is 62.4 lbs per cubic foot.

Set up and evaluate the integral that gives the volume of the solid formed by revolving the region bounded by $y = x ^ { 8 }$ and $y = 256$ in the first quadrant about the $y$ -axis.

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the given lines. $y = 22 - 10 x - x ^ { 2 } , y = x + 22$ (i) $x$ -axis; (ii) the line $y = 11$

Use the disk or the shell method to find the volume of the solid generated by revolving the region bounded by the graphs of the equations $y = \frac { 17 } { x ^ { 2 } } , y = 0 , x = 1 , x = 7$ about the $x$ -axis. Round your answer to two decimal places.

Find the area of the region bounded by the equations by integrating (i) with respect to x and (ii) with respect to y. x=16- x=y-4

Find the area of the region bounded by the graphs of the equations. $f ( x ) = \sin ( x ) , g ( x ) = \cos ( 2 x ) , \frac { - \pi } { 2 } \leq x \leq \frac { \pi } { 6 }$

A circular plate of radius r feet is submerged vertically in a tank of fluid that weighs $w$ pounds per cubic foot. The center of the circle is $k ( k > r )$ feet below the surface of the fluid. The fluid force on the surface of the plate is given by $F = w k \left( \pi r ^ { 2 } \right)$ Find the fluid force on the circular plate as shown in the figure given $a = 5$ feet and $b = 2$ feet. Round your answer to one decimal place.

Use the disk or shell method to find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the line $x = 22$ . $y = x ^ { 3 } , y = 0 , x = 11$

Find the area of the region bounded by the graphs of the function $f ( x ) = \sin 5 x$ $g ( x ) = \cos 10 x , \frac { - \pi } { 10 } \leq x \leq \frac { \pi } { 30 }$ . Round your answer to three decimal places.

Set up and evaluate integrals for finding the moment about the y-axis for the region bounded by the graphs of the equations. (Assume $\rho = 1 .$ ) $y = 64 - x ^ { 2 } , y = 0$

Use the shell method to set up and evaluate an integral that gives the volume of the solid generated by revolving the plane region about the y-axis. $y = 25 - x ^ { 2 } , y = 0$

Find the buoyant force of a rectangular solid of the given dimensions submerged in water so that the top side is parallel to the surface of the water. The buoyant force is the difference Between the fluid forces on the top and bottom sides of the solid. Round your answer to two decimal Places.