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1. Final Exam
1. Find the area bounded by the curve 𝑦 = 3𝑥(𝑥 − 2)2
and the
x axis.
2. The area bounded by the hyperbola 𝑥2
− 𝑦2
= 𝑎2
, the x-
axis, and the line x= 2a revolves Ox (x axis). Find the volume.
(Used circular disk)
3. The area bounded by the parabola 𝑎𝑦 = 𝑥2
, the x- axis, the
line x= b is revolved about the y-axis. Find the volume
generated. (Used cylindrical shell method)
Final Exam
1. Find the area bounded by the curve 𝑦 = 3𝑥(𝑥 − 2)2
and the
x axis.
2. The area bounded by the hyperbola 𝑥2
− 𝑦2
= 𝑎2
, the x-
axis, and the line x= 2a revolves Ox (x axis). Find the volume.
(Used circular disk)
3. The area bounded by the parabola 𝑎𝑦 = 𝑥2
, the x- axis, the
line x= b is revolved about the y-axis. Find the volume
generated. (Used cylindrical shell method)
Final Exam
1. Find the area bounded by the curve 𝑦 = 3𝑥(𝑥 − 2)2
and the
x axis.
2. The area bounded by the hyperbola 𝑥2
− 𝑦2
= 𝑎2
, the x-
axis, and the line x= 2a revolves Ox (x axis). Find the volume.
(Used circular disk)
3. The area bounded by the parabola 𝑎𝑦 = 𝑥2
, the x- axis, the
line x= b is revolved about the y-axis. Find the volume
generated. (Used cylindrical shell method)
Final Exam
1. Find the area bounded by the curve 𝑦 = 3𝑥(𝑥 − 2)2
and the
x axis.
2. The area bounded by the hyperbola 𝑥2
− 𝑦2
= 𝑎2
, the x-
axis, and the line x= 2a revolves Ox (x axis). Find the volume.
(Used circular disk)
3. The area bounded by the parabola 𝑎𝑦 = 𝑥2
, the x- axis, the
line x= b is revolved about the y-axis. Find the volume
generated. (Used cylindrical shell method)
Final Exam
1. Find the area bounded by the curve 𝑦 = 3𝑥(𝑥 − 2)2
and the
x axis.
2. The area bounded by the hyperbola 𝑥2
− 𝑦2
= 𝑎2
, the x-
axis, and the line x= 2a revolves Ox (x axis). Find the volume.
(Used circular disk)
3. The area bounded by the parabola 𝑎𝑦 = 𝑥2
, the x- axis, the
line x= b is revolved about the y-axis. Find the volume
generated. (Used cylindrical shell method)
Final Exam
1. Find the area bounded by the curve 𝑦 = 3𝑥(𝑥 − 2)2
and the
x axis.
2. The area bounded by the hyperbola 𝑥2
− 𝑦2
= 𝑎2
, the x-
axis, and the line x= 2a revolves Ox (x axis). Find the volume.
(Used circular disk)
3. The area bounded by the parabola 𝑎𝑦 = 𝑥2
, the x- axis, the
line x= b is revolved about the y-axis. Find the volume
generated. (Used cylindrical shell method)
Final Exam
1. Find the area bounded by the curve 𝑦 = 3𝑥(𝑥 − 2)2
and the
x axis.
2. The area bounded by the hyperbola 𝑥2
− 𝑦2
= 𝑎2
, the x-
axis, and the line x= 2a revolves Ox (x axis). Find the volume.
(Used circular disk)
3. The area bounded by the parabola 𝑎𝑦 = 𝑥2
, the x- axis, the
line x= b is revolved about the y-axis. Find the volume
generated. (Used cylindrical shell method)
Final Exam
1. Find the area bounded by the curve 𝑦 = 3𝑥(𝑥 − 2)2
and the
x axis.
2. The area bounded by the hyperbola 𝑥2
− 𝑦2
= 𝑎2
, the x-
axis, and the line x= 2a revolves Ox (x axis). Find the volume.
(Used circular disk)
3. The area bounded by the parabola 𝑎𝑦 = 𝑥2
, the x- axis, the
line x= b is revolved about the y-axis. Find the volume
generated. (Used cylindrical shell method)
Final Exam
1. Find the area bounded by the curve 𝑦 = 3𝑥(𝑥 − 2)2
and the
x axis.
2. The area bounded by the hyperbola 𝑥2
− 𝑦2
= 𝑎2
, the x-
axis, and the line x= 2a revolves Ox (x axis). Find the volume.
(Used circular disk)
3. The area bounded by the parabola 𝑎𝑦 = 𝑥2
, the x- axis, the
line x= b is revolved about the y-axis. Find the volume
generated. (Used cylindrical shell method)
Final Exam
1. Find the area bounded by the curve 𝑦 = 3𝑥(𝑥 − 2)2
and the
x axis.
2. The area bounded by the hyperbola 𝑥2
− 𝑦2
= 𝑎2
, the x-
axis, and the line x= 2a revolves Ox (x axis). Find the volume.
(Used circular disk)
3. The area bounded by the parabola 𝑎𝑦 = 𝑥2
, the x- axis, the
line x= b is revolved about the y-axis. Find the volume
generated. (Used cylindrical shell method)