Integrate with respect to x

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In summary, integration with respect to x involves finding the area under a curve in the x-direction by adding up infinitely small rectangles. It differs from integration with respect to y in the direction of measurement. Its purpose is to solve problems involving rates of change, and it is the inverse operation of differentiation. Real-life applications of integration with respect to x include physics, engineering, economics, and statistics for finding areas, volumes, and averages and solving problems involving rates of change.
  • #1
cabellos
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I just need a check on when integrating sin(xy) with respect to x? Does this become -cos(xy) or -cos(x)

thanks
 
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  • #2
Neither. y is to be treated as a constant. And from that follows?
 
  • #3
ok....is it -cos(xy) / y
 
  • #4
yes that's correct.
 

1. What does it mean to integrate with respect to x?

Integrating with respect to x means finding the area under a curve in the x-direction. It involves adding up all the infinitely small rectangles that make up the curve.

2. How is integration with respect to x different from integration with respect to y?

The main difference is the direction in which the area is being measured. Integrating with respect to x measures the area in the x-direction, while integrating with respect to y measures the area in the y-direction.

3. What is the purpose of integrating with respect to x?

Integrating with respect to x is used to solve problems involving rates of change, such as finding the distance traveled by an object with varying velocity or the volume of a three-dimensional shape with varying cross-sectional area.

4. How is integration with respect to x related to derivatives?

Integration with respect to x is the inverse operation of differentiation, which is finding the rate of change of a function. Integration allows us to recover the original function from its derivative by finding the area under the curve of the derivative.

5. Are there any applications of integration with respect to x in real life?

Integrating with respect to x has many real-life applications, such as in physics, engineering, economics, and statistics. It is used to solve problems involving rates of change and to find areas, volumes, and averages in various contexts.

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