Integral of the inverse to $f(x)$
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The formula $$f(x)=sin(x)+frac{2x}{pi}$$ defines the function$ f:[0, frac{pi}{2}] rightarrow [0, 2] $.
How should you go about finding: $int_{0}^{2}f^{-1}(y)dy$?
calculus integration inverse-function
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TheSimpliFire
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up vote
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favorite
The formula $$f(x)=sin(x)+frac{2x}{pi}$$ defines the function$ f:[0, frac{pi}{2}] rightarrow [0, 2] $.
How should you go about finding: $int_{0}^{2}f^{-1}(y)dy$?
calculus integration inverse-function
edited yesterday
TheSimpliFire
11.6k62256
asked yesterday
Curl
6012
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
The formula $$f(x)=sin(x)+frac{2x}{pi}$$ defines the function$ f:[0, frac{pi}{2}] rightarrow [0, 2] $.
How should you go about finding: $int_{0}^{2}f^{-1}(y)dy$?
calculus integration inverse-function
edited yesterday
TheSimpliFire
11.6k62256
asked yesterday
Curl
6012
The formula $$f(x)=sin(x)+frac{2x}{pi}$$ defines the function$ f:[0, frac{pi}{2}] rightarrow [0, 2] $.
How should you go about finding: $int_{0}^{2}f^{-1}(y)dy$?
calculus integration inverse-function
calculus integration inverse-function
edited yesterday
TheSimpliFire
11.6k62256
asked yesterday
Curl
6012
edited yesterday
TheSimpliFire
11.6k62256
asked yesterday
Curl
6012
edited yesterday
TheSimpliFire
11.6k62256
edited yesterday
TheSimpliFire
11.6k62256
edited yesterday
TheSimpliFire
11.6k62256
11.6k62256
asked yesterday
Curl
6012
asked yesterday
Curl
6012
asked yesterday
Curl
6012
6012
add a comment |
add a comment |
3 Answers
3
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up vote
6
down vote
Hint:
First use substitution $y=f(x)$
$$int_{0}^{2}f^{-1}(y)dy=int_{f^{-1}(0)}^{f^{-1}(2)}xf'(x) dx$$
and then apply integration by parts.
answered yesterday
Nosrati
25.8k62252
It's important to note that one should check whether the function is monotonic over the interval before applying this procedure.
– Acccumulation
yesterday
Yeah, A good point!
– Nosrati
yesterday
add a comment |
up vote
6
down vote
Hint: Think geometrically. That integral represents some area in the plane. You can easily calculate the area of a certain rectangle containing that area, and with a little effort the remaining part of the rectangle.
answered yesterday
Arthur
108k7103186
add a comment |
up vote
0
down vote
Use substitution $y=f(x)$ and integrate by parts.
answered yesterday
Loara
1
New contributor
Loara is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
This would be more useful if you showed the steps to carry out this program. Details like the change in limits of integration are important here (for the definite integral).
– hardmath
yesterday
add a comment |
3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
6
down vote
Hint:
First use substitution $y=f(x)$
$$int_{0}^{2}f^{-1}(y)dy=int_{f^{-1}(0)}^{f^{-1}(2)}xf'(x) dx$$
and then apply integration by parts.
answered yesterday
Nosrati
25.8k62252
It's important to note that one should check whether the function is monotonic over the interval before applying this procedure.
– Acccumulation
yesterday
Yeah, A good point!
– Nosrati
yesterday
add a comment |
up vote
6
down vote
Hint:
First use substitution $y=f(x)$
$$int_{0}^{2}f^{-1}(y)dy=int_{f^{-1}(0)}^{f^{-1}(2)}xf'(x) dx$$
and then apply integration by parts.
answered yesterday
Nosrati
25.8k62252
It's important to note that one should check whether the function is monotonic over the interval before applying this procedure.
– Acccumulation
yesterday
Yeah, A good point!
– Nosrati
yesterday
add a comment |
up vote
6
down vote
up vote
6
down vote
Hint:
First use substitution $y=f(x)$
$$int_{0}^{2}f^{-1}(y)dy=int_{f^{-1}(0)}^{f^{-1}(2)}xf'(x) dx$$
and then apply integration by parts.
answered yesterday
Nosrati
25.8k62252
Hint:
First use substitution $y=f(x)$
$$int_{0}^{2}f^{-1}(y)dy=int_{f^{-1}(0)}^{f^{-1}(2)}xf'(x) dx$$
and then apply integration by parts.
answered yesterday
Nosrati
25.8k62252
answered yesterday
Nosrati
25.8k62252
answered yesterday
Nosrati
25.8k62252
answered yesterday
Nosrati
25.8k62252
25.8k62252
It's important to note that one should check whether the function is monotonic over the interval before applying this procedure.
– Acccumulation
yesterday
Yeah, A good point!
– Nosrati
yesterday
add a comment |
It's important to note that one should check whether the function is monotonic over the interval before applying this procedure.
– Acccumulation
yesterday
Yeah, A good point!
– Nosrati
yesterday
It's important to note that one should check whether the function is monotonic over the interval before applying this procedure.
– Acccumulation
yesterday
It's important to note that one should check whether the function is monotonic over the interval before applying this procedure.
– Acccumulation
yesterday
Yeah, A good point!
– Nosrati
yesterday
Yeah, A good point!
– Nosrati
yesterday
add a comment |
up vote
6
down vote
Hint: Think geometrically. That integral represents some area in the plane. You can easily calculate the area of a certain rectangle containing that area, and with a little effort the remaining part of the rectangle.
answered yesterday
Arthur
108k7103186
add a comment |
up vote
6
down vote
Hint: Think geometrically. That integral represents some area in the plane. You can easily calculate the area of a certain rectangle containing that area, and with a little effort the remaining part of the rectangle.
answered yesterday
Arthur
108k7103186
add a comment |
up vote
6
down vote
up vote
6
down vote
Hint: Think geometrically. That integral represents some area in the plane. You can easily calculate the area of a certain rectangle containing that area, and with a little effort the remaining part of the rectangle.
answered yesterday
Arthur
108k7103186
Hint: Think geometrically. That integral represents some area in the plane. You can easily calculate the area of a certain rectangle containing that area, and with a little effort the remaining part of the rectangle.
answered yesterday
Arthur
108k7103186
edited yesterday
answered yesterday
Arthur
108k7103186
answered yesterday
Arthur
108k7103186
answered yesterday
Arthur
108k7103186
108k7103186
add a comment |
add a comment |
up vote
0
down vote
Use substitution $y=f(x)$ and integrate by parts.
answered yesterday
Loara
1
New contributor
Loara is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
This would be more useful if you showed the steps to carry out this program. Details like the change in limits of integration are important here (for the definite integral).
– hardmath
yesterday
add a comment |
up vote
0
down vote
Use substitution $y=f(x)$ and integrate by parts.
answered yesterday
Loara
1
New contributor
Loara is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
This would be more useful if you showed the steps to carry out this program. Details like the change in limits of integration are important here (for the definite integral).
– hardmath
yesterday
add a comment |
up vote
0
down vote
up vote
0
down vote
Use substitution $y=f(x)$ and integrate by parts.
answered yesterday
Loara
1
New contributor
Loara is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Use substitution $y=f(x)$ and integrate by parts.
answered yesterday
Loara
1
New contributor
Loara is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
answered yesterday
Loara
1
New contributor
Loara is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
answered yesterday
Loara
1
answered yesterday
Loara
1
1
New contributor
Loara is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
Loara is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Loara is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
This would be more useful if you showed the steps to carry out this program. Details like the change in limits of integration are important here (for the definite integral).
– hardmath
yesterday
add a comment |
This would be more useful if you showed the steps to carry out this program. Details like the change in limits of integration are important here (for the definite integral).
– hardmath
yesterday
This would be more useful if you showed the steps to carry out this program. Details like the change in limits of integration are important here (for the definite integral).
– hardmath
yesterday
This would be more useful if you showed the steps to carry out this program. Details like the change in limits of integration are important here (for the definite integral).
– hardmath
yesterday
add a comment |
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