Question text 4Marks Calculate the area of the region bounded by the curves with equations [math: f(x)=x24−2] and [math: g(x)=x+1] Your answer will be of the form [math: p/q] where [math: p] and [math: q] are positive integers with no common factors. Enter [math: p] and [math: q]. [math: p=] Answer 3[input] and [math: q=] Answer 4[input]. Notes Report question issue Question 8 Notes多项填空题
登录即可查看完整答案
我们收录了全球超50000道真实原题与详细解析,现在登录,立即获得答案。

Question

Question text 4Marks Calculate the area of the region bounded by the curves with equations [math: f(x)=x24−2] and [math: g(x)=x+1] Your answer will be of the form [math: p/q] where [math: p] and [math: q] are positive integers with no common factors. Enter [math: p] and [math: q]. [math: p=] Answer 3[input] and [math: q=] Answer 4[input]. Notes Report question issue Question 8 Notes多项填空题

登录即可查看完整答案

我们收录了全球超50000道真实原题与详细解析,现在登录,立即获得答案。

BlackTom AI · Accepted Answer

Set the two curves equal to find the intersection points: x^2/4 − 2 = x + 1. Multiply by 4: x^2 − 8 = 4x + 4, so x^2 − 4x − 12 = 0. Factor: (x − 6)(x + 2) = 0, so the curves meet at x = −2 and x = 6. These are the x–bounds of the bounded region.

Determine which curve is on top between the intersections. Evaluate at x = 0: f(0) = 0^2/4 − 2 = −2, and g(0) = 0 + 1 = 1, so g(x) = x + 1 lies above f(x) = x^2/4 − 2 on the interval [−2, 6]. Therefore the area A is the integral from −2 to 6 of (g(x) − f(x)) dx.

Compute the integrand:
g(x) − f(x) = (x + 1) − (x^2/4 − 2) = −(x^2)/4 + x + 3.

Find an antiderivative:
∫(−x^2/4 + x + 3) dx = −x^3/12 + x^2/2 + 3x + C.

Evaluate at the bounds:
At x = 6: F(6) = −6^3/12 + 6^2/2 + 3·6 = −216/12 + 36/2 + 18 = −18 + 18 + 18 = 18.
At x = −2: F(−2) = −(−2)^3/12 + (−2)^2/2 + 3(−2) = +8/12 + 4/2 − 6 = 2/3 + 2 − 6 = 2/3 − 4 = −10/3.

Subtract to get the area:
A = F(6) − F(−2) = 18 − (−10/3) = 18 + 10/3 = (54 + 10)/3 = 64/3.

Thus the area of the region bounded by f(x) = x^2/4 − 2 and g(x) = x + 1 is 64/3. Writing this as p/q with p and q positive integers in lowest terms gives p = 64 and q = 3.

Why the given answer p = 19, q = 6 would be incorrect: the intersection points and definite integral above are straightforward algebra and calculus; they produce 64/3, not 19/6, so 19/6 does not match the computed exact area. Conclusion: the correct computed area is 64/3 (p = 64, q = 3), which differs from the provided answer (19/6).

类似问题

Which one of the following is equal to the shaded area above?

The area bounded by the two graphs above is equal to:

Find the shaded area above.

Which one of the following is equal to the shaded area above?

Find the exact shaded area above.

Find the area of the region bounded by the graphs of the function , , . Round your answer to three decimal places.

Find the area of the region bounded by the graphs of the algebraic functions.

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

更多留学生实用工具

智能学习助手

风格化写作助手

论文查重助手

文献引用助手

课堂转译助手

课堂笔记助手

Quiz搜索助手

学校历年真题

智能刷题助手

智能匹配练习

希望你的学习变得更简单

类似问题

Which one of the following is equal to the shaded area above?

The area bounded by the two graphs above is equal to:

Find the shaded area above.

Which one of the following is equal to the shaded area above?

Find the exact shaded area above.

Find the area of the region bounded by the graphs of the function , , . Round your answer to three decimal places. ​

Find the area of the region bounded by the graphs of the algebraic functions. ​ ​

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

更多留学生实用工具

希望你的学习变得更简单

Find the area of the region bounded by the graphs of the function , , . Round your answer to three decimal places.

Find the area of the region bounded by the graphs of the algebraic functions.

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