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Grade 12 · Substitution and bounds

Substitution and bounds: A gentle exponent

Substitution and bounds: investigate a gentle exponent with exponent factor k = 0.1; final upper bound b = 1.

Predict what will happen, press Play, then pause and explain what changed. Every control also works without playback.

Watch the relationship

Paused
Substitution and bounds: A gentle exponent. Upper x bound: 0. Transformed u bound: 0. Exact integral: 0The upper bound changes with the variable00.2200.51yx → · labeled axes rescale to this model
k and the final bound are positive, making the substitution increasing on the chosen interval. The result is an exact antiderivative evaluation. Axis rescaling is used because exponential values vary strongly across the controls.

Starts paused. Play once, pause anywhere, or use Step to inspect the mathematics. Playback stops when this panel leaves the screen.

Make it your experiment

Change one value. Notice what follows.

The controls adjust the model. Numbers below describe the current frame. Decimals are rounded.

Upper x bound
0
Transformed u bound
0
Exact integral
0

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Understand what you are seeing

The idea behind the motion.

Substitution replaces a composite expression and its differential together. Here 2kx dx is exactly du when u=kx². Moving the upper x-bound also moves the upper u-bound, so retaining the original bound after changing variables would integrate over the wrong interval. This investigation starts with Exponent factor k = 0.1; Final upper bound B = 1. Predict the result before playing, then change one parameter while keeping the other fixed to test the reason for the change.

A relationship to keep

∫₀ᵀ2kx e^(kx²)dx=e^(kT²)−1; u=kx²

Read the symbols alongside the explanation. A diagram shows the relationship; the assumptions tell you when it applies.

  1. STEP 1

    Set up the mathematical model

    Identify the inner expression kx² and its derivative 2kx. The starting case is “A gentle exponent.”

  2. STEP 2

    Follow the changing quantity

    Expand the x-interval from zero to the selected endpoint. Track the corresponding upper u-bound kT².

  3. STEP 3

    Explain and test the result

    Evaluate e raised to the transformed bound minus one. Differentiate that accumulation with respect to T to recover the original integrand.

Your turn to explain

Make a prediction. Test your reasoning.

Keep Exponent factor k = 0.1; Final upper bound B = 1. Pause the timeline at 40%. Given upper x bound = 0.4, calculate transformed u bound, exact integral. Show the substitution into the displayed formula.

Use the values specified in the question. Reset restores the initial values for this investigation.

Compare your explanation

Under u=kx², the upper bound becomes (0.1)(0.4)²=0.016. Therefore ∫eᵘdu from 0 to 0.016 equals e^0.016−1=0.016129. Results: Transformed u bound: 0.016; Exact integral: 0.016. Decimal values are rounded; retain the original parameters when checking.

Work through a full lesson

Connect the animation to a worked example and practice questions.

Grade 12 · Calculus

Change integral limits along with the variable

Use a substitution consistently in a definite integral.